Simplify: ${ \left(\frac{x A}{x B}\right) {a 2+ab+b^2} \times \left(\frac{x {b 2+bc}}{x {-c 2}}\right)^{b-c} \times \left(\frac{x {c 2}}{x {-ca-a 2}}\right)^{c-a} }$

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Introduction

In this article, we will delve into the world of algebra and simplify a complex expression involving exponents and fractions. The given expression is a product of three terms, each containing exponents and fractions. Our goal is to simplify this expression and provide a clear understanding of the underlying mathematical concepts.

The Given Expression

The given expression is:

(xaxb)a2+ab+b2Γ—(xb2+bcxβˆ’c2)bβˆ’cΓ—(xc2xβˆ’caβˆ’a2)cβˆ’a\left(\frac{x^a}{x^b}\right)^{a^2+ab+b^2} \times \left(\frac{x^{b^2+bc}}{x^{-c^2}}\right)^{b-c} \times \left(\frac{x^{c^2}}{x^{-ca-a^2}}\right)^{c-a}

Simplifying the First Term

Let's start by simplifying the first term:

(xaxb)a2+ab+b2\left(\frac{x^a}{x^b}\right)^{a^2+ab+b^2}

Using the property of exponents that states (xa)b=xab(x^a)^b = x^{ab}, we can rewrite the first term as:

xa(a2+ab+b2)xb(a2+ab+b2)\frac{x^{a(a^2+ab+b^2)}}{x^{b(a^2+ab+b^2)}}

Simplifying the Second Term

Next, let's simplify the second term:

(xb2+bcxβˆ’c2)bβˆ’c\left(\frac{x^{b^2+bc}}{x^{-c^2}}\right)^{b-c}

Using the property of exponents that states (xa)b=xab(x^a)^b = x^{ab}, we can rewrite the second term as:

x(b2+bc)(bβˆ’c)x(βˆ’c2)(bβˆ’c)\frac{x^{(b^2+bc)(b-c)}}{x^{(-c^2)(b-c)}}

Simplifying the Third Term

Finally, let's simplify the third term:

(xc2xβˆ’caβˆ’a2)cβˆ’a\left(\frac{x^{c^2}}{x^{-ca-a^2}}\right)^{c-a}

Using the property of exponents that states (xa)b=xab(x^a)^b = x^{ab}, we can rewrite the third term as:

x(c2)(cβˆ’a)x(βˆ’caβˆ’a2)(cβˆ’a)\frac{x^{(c^2)(c-a)}}{x^{(-ca-a^2)(c-a)}}

Combining the Terms

Now that we have simplified each term, let's combine them:

xa(a2+ab+b2)xb(a2+ab+b2)Γ—x(b2+bc)(bβˆ’c)x(βˆ’c2)(bβˆ’c)Γ—x(c2)(cβˆ’a)x(βˆ’caβˆ’a2)(cβˆ’a)\frac{x^{a(a^2+ab+b^2)}}{x^{b(a^2+ab+b^2)}} \times \frac{x^{(b^2+bc)(b-c)}}{x^{(-c^2)(b-c)}} \times \frac{x^{(c^2)(c-a)}}{x^{(-ca-a^2)(c-a)}}

Canceling Out Common Factors

We can simplify the expression further by canceling out common factors:

xa(a2+ab+b2)+(b2+bc)(bβˆ’c)+(c2)(cβˆ’a)xb(a2+ab+b2)+(βˆ’c2)(bβˆ’c)+(βˆ’caβˆ’a2)(cβˆ’a)\frac{x^{a(a^2+ab+b^2) + (b^2+bc)(b-c) + (c^2)(c-a)}}{x^{b(a^2+ab+b^2) + (-c^2)(b-c) + (-ca-a^2)(c-a)}}

Simplifying the Exponents

Let's simplify the exponents:

a(a2+ab+b2)+(b2+bc)(bβˆ’c)+(c2)(cβˆ’a)a(a^2+ab+b^2) + (b^2+bc)(b-c) + (c^2)(c-a)

=a3+ab2+b3+b3βˆ’bc2+bc2+c3βˆ’ac2βˆ’a3= a^3 + ab^2 + b^3 + b^3 - bc^2 + bc^2 + c^3 - ac^2 - a^3

=b3+c3βˆ’ac2= b^3 + c^3 - ac^2

b(a2+ab+b2)+(βˆ’c2)(bβˆ’c)+(βˆ’caβˆ’a2)(cβˆ’a)b(a^2+ab+b^2) + (-c^2)(b-c) + (-ca-a^2)(c-a)

=ab2+ab3+b3βˆ’bc2+bc2βˆ’c3a+ca2βˆ’ca3+a2c2= ab^2 + ab^3 + b^3 - bc^2 + bc^2 - c^3a + ca^2 - ca^3 + a^2c^2

=ab3+b3+a2c2βˆ’c3a= ab^3 + b^3 + a^2c^2 - c^3a

Canceling Out Common Factors

We can simplify the expression further by canceling out common factors:

xb3+c3βˆ’ac2xab3+b3+a2c2βˆ’c3a\frac{x^{b^3 + c^3 - ac^2}}{x^{ab^3 + b^3 + a^2c^2 - c^3a}}

Final Simplification

We can simplify the expression further by canceling out common factors:

xb3+c3βˆ’ac2βˆ’ab3βˆ’b3βˆ’a2c2+c3ax^{b^3 + c^3 - ac^2 - ab^3 - b^3 - a^2c^2 + c^3a}

=xc3βˆ’ac2+c3aβˆ’a2c2βˆ’b3= x^{c^3 - ac^2 + c^3a - a^2c^2 - b^3}

=xc3(1βˆ’a+a)βˆ’a2c2βˆ’b3= x^{c^3(1 - a + a) - a^2c^2 - b^3}

=xc3βˆ’a2c2βˆ’b3= x^{c^3 - a^2c^2 - b^3}

Conclusion

In this article, we simplified a complex algebraic expression involving exponents and fractions. We used the properties of exponents to simplify each term and then combined them. Finally, we canceled out common factors and simplified the expression further. The final simplified expression is:

xc3βˆ’a2c2βˆ’b3x^{c^3 - a^2c^2 - b^3}

This expression can be further simplified by factoring out common terms, but for the purpose of this article, we have reached the final simplified form.

Final Answer

The final simplified expression is:

x^{c^3 - a^2c^2 - b^3}$<br/> # Simplify: A Complex Algebraic Expression - Q&A =====================================================

Introduction

In our previous article, we simplified a complex algebraic expression involving exponents and fractions. We used the properties of exponents to simplify each term and then combined them. In this article, we will answer some frequently asked questions related to the simplification of the expression.

Q&A

Q: What is the final simplified expression?

A: The final simplified expression is:

xc3βˆ’a2c2βˆ’b3</span></p><h3>Q:Howdidyousimplifytheexpression?</h3><p>A:Weusedthepropertiesofexponentstosimplifyeachtermandthencombinedthem.Wealsocanceledoutcommonfactorstosimplifytheexpressionfurther.</p><h3>Q:Whatarethepropertiesofexponentsthatyouused?</h3><p>A:Weusedthefollowingpropertiesofexponents:</p><ul><li><spanclass="katex"><spanclass="katexβˆ’mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mostretchy="false">(</mo><msup><mi>x</mi><mi>a</mi></msup><msup><mostretchy="false">)</mo><mi>b</mi></msup><mo>=</mo><msup><mi>x</mi><mrow><mi>a</mi><mi>b</mi></mrow></msup></mrow><annotationencoding="application/xβˆ’tex">(xa)b=xab</annotation></semantics></math></span><spanclass="katexβˆ’html"ariaβˆ’hidden="true"><spanclass="base"><spanclass="strut"style="height:1.0991em;verticalβˆ’align:βˆ’0.25em;"></span><spanclass="mopen">(</span><spanclass="mord"><spanclass="mordmathnormal">x</span><spanclass="msupsub"><spanclass="vlistβˆ’t"><spanclass="vlistβˆ’r"><spanclass="vlist"style="height:0.6644em;"><spanstyle="top:βˆ’3.063em;marginβˆ’right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingresetβˆ’size6size3mtight"><spanclass="mordmathnormalmtight">a</span></span></span></span></span></span></span></span><spanclass="mclose"><spanclass="mclose">)</span><spanclass="msupsub"><spanclass="vlistβˆ’t"><spanclass="vlistβˆ’r"><spanclass="vlist"style="height:0.8491em;"><spanstyle="top:βˆ’3.063em;marginβˆ’right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingresetβˆ’size6size3mtight"><spanclass="mordmathnormalmtight">b</span></span></span></span></span></span></span></span><spanclass="mspace"style="marginβˆ’right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="marginβˆ’right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.8491em;"></span><spanclass="mord"><spanclass="mordmathnormal">x</span><spanclass="msupsub"><spanclass="vlistβˆ’t"><spanclass="vlistβˆ’r"><spanclass="vlist"style="height:0.8491em;"><spanstyle="top:βˆ’3.063em;marginβˆ’right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingresetβˆ’size6size3mtight"><spanclass="mordmtight"><spanclass="mordmathnormalmtight">ab</span></span></span></span></span></span></span></span></span></span></span></span></li><li><spanclass="katex"><spanclass="katexβˆ’mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><msup><mi>x</mi><mi>a</mi></msup><msup><mi>x</mi><mi>b</mi></msup></mfrac><mo>=</mo><msup><mi>x</mi><mrow><mi>a</mi><mo>βˆ’</mo><mi>b</mi></mrow></msup></mrow><annotationencoding="application/xβˆ’tex">xaxb=xaβˆ’b</annotation></semantics></math></span><spanclass="katexβˆ’html"ariaβˆ’hidden="true"><spanclass="base"><spanclass="strut"style="height:1.2684em;verticalβˆ’align:βˆ’0.3574em;"></span><spanclass="mord"><spanclass="mopennulldelimiter"></span><spanclass="mfrac"><spanclass="vlistβˆ’tvlistβˆ’t2"><spanclass="vlistβˆ’r"><spanclass="vlist"style="height:0.911em;"><spanstyle="top:βˆ’2.6426em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="sizingresetβˆ’size6size3mtight"><spanclass="mordmtight"><spanclass="mordmtight"><spanclass="mordmathnormalmtight">x</span><spanclass="msupsub"><spanclass="vlistβˆ’t"><spanclass="vlistβˆ’r"><spanclass="vlist"style="height:0.782em;"><spanstyle="top:βˆ’2.786em;marginβˆ’right:0.0714em;"><spanclass="pstrut"style="height:2.5em;"></span><spanclass="sizingresetβˆ’size3size1mtight"><spanclass="mordmathnormalmtight">b</span></span></span></span></span></span></span></span></span></span></span><spanstyle="top:βˆ’3.23em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="fracβˆ’line"style="borderβˆ’bottomβˆ’width:0.04em;"></span></span><spanstyle="top:βˆ’3.394em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="sizingresetβˆ’size6size3mtight"><spanclass="mordmtight"><spanclass="mordmtight"><spanclass="mordmathnormalmtight">x</span><spanclass="msupsub"><spanclass="vlistβˆ’t"><spanclass="vlistβˆ’r"><spanclass="vlist"style="height:0.7385em;"><spanstyle="top:βˆ’2.931em;marginβˆ’right:0.0714em;"><spanclass="pstrut"style="height:2.5em;"></span><spanclass="sizingresetβˆ’size3size1mtight"><spanclass="mordmathnormalmtight">a</span></span></span></span></span></span></span></span></span></span></span></span><spanclass="vlistβˆ’s">​</span></span><spanclass="vlistβˆ’r"><spanclass="vlist"style="height:0.3574em;"><span></span></span></span></span></span><spanclass="mclosenulldelimiter"></span></span><spanclass="mspace"style="marginβˆ’right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="marginβˆ’right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.8491em;"></span><spanclass="mord"><spanclass="mordmathnormal">x</span><spanclass="msupsub"><spanclass="vlistβˆ’t"><spanclass="vlistβˆ’r"><spanclass="vlist"style="height:0.8491em;"><spanstyle="top:βˆ’3.063em;marginβˆ’right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingresetβˆ’size6size3mtight"><spanclass="mordmtight"><spanclass="mordmathnormalmtight">a</span><spanclass="mbinmtight">βˆ’</span><spanclass="mordmathnormalmtight">b</span></span></span></span></span></span></span></span></span></span></span></span></li></ul><h3>Q:Canyouexplainthesimplificationprocessinmoredetail?</h3><p>A:Hereisastepβˆ’byβˆ’stepexplanationofthesimplificationprocess:</p><ol><li>Simplifyeachtermusingthepropertiesofexponents.</li><li>Combinethesimplifiedterms.</li><li>Canceloutcommonfactors.</li><li>Simplifytheexpressionfurther.</li></ol><h3>Q:Whatisthesignificanceoftheexpression?</h3><p>A:Theexpressionisaproductofthreeterms,eachcontainingexponentsandfractions.Theexpressioncanbeusedtomodelrealβˆ’worldproblemsinvolvingexponentialgrowthanddecay.</p><h3>Q:Canyouprovidemoreexamplesofsimplifyingexpressionsinvolvingexponentsandfractions?</h3><p>A:Yes,hereareafewexamples:</p><ul><li><spanclass="katex"><spanclass="katexβˆ’mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mrow><mofence="true">(</mo><mfrac><msup><mi>x</mi><mi>a</mi></msup><msup><mi>x</mi><mi>b</mi></msup></mfrac><mofence="true">)</mo></mrow><mi>c</mi></msup><mo>=</mo><mfrac><msup><mi>x</mi><mrow><mi>a</mi><mi>c</mi></mrow></msup><msup><mi>x</mi><mrow><mi>b</mi><mi>c</mi></mrow></msup></mfrac></mrow><annotationencoding="application/xβˆ’tex">(xaxb)c=xacxbc</annotation></semantics></math></span><spanclass="katexβˆ’html"ariaβˆ’hidden="true"><spanclass="base"><spanclass="strut"style="height:1.3227em;verticalβˆ’align:βˆ’0.3574em;"></span><spanclass="minner"><spanclass="minner"><spanclass="mopendelimcenter"style="top:0em;"><spanclass="delimsizingsize1">(</span></span><spanclass="mord"><spanclass="mopennulldelimiter"></span><spanclass="mfrac"><spanclass="vlistβˆ’tvlistβˆ’t2"><spanclass="vlistβˆ’r"><spanclass="vlist"style="height:0.911em;"><spanstyle="top:βˆ’2.6426em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="sizingresetβˆ’size6size3mtight"><spanclass="mordmtight"><spanclass="mordmtight"><spanclass="mordmathnormalmtight">x</span><spanclass="msupsub"><spanclass="vlistβˆ’t"><spanclass="vlistβˆ’r"><spanclass="vlist"style="height:0.782em;"><spanstyle="top:βˆ’2.786em;marginβˆ’right:0.0714em;"><spanclass="pstrut"style="height:2.5em;"></span><spanclass="sizingresetβˆ’size3size1mtight"><spanclass="mordmathnormalmtight">b</span></span></span></span></span></span></span></span></span></span></span><spanstyle="top:βˆ’3.23em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="fracβˆ’line"style="borderβˆ’bottomβˆ’width:0.04em;"></span></span><spanstyle="top:βˆ’3.394em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="sizingresetβˆ’size6size3mtight"><spanclass="mordmtight"><spanclass="mordmtight"><spanclass="mordmathnormalmtight">x</span><spanclass="msupsub"><spanclass="vlistβˆ’t"><spanclass="vlistβˆ’r"><spanclass="vlist"style="height:0.7385em;"><spanstyle="top:βˆ’2.931em;marginβˆ’right:0.0714em;"><spanclass="pstrut"style="height:2.5em;"></span><spanclass="sizingresetβˆ’size3size1mtight"><spanclass="mordmathnormalmtight">a</span></span></span></span></span></span></span></span></span></span></span></span><spanclass="vlistβˆ’s">​</span></span><spanclass="vlistβˆ’r"><spanclass="vlist"style="height:0.3574em;"><span></span></span></span></span></span><spanclass="mclosenulldelimiter"></span></span><spanclass="mclosedelimcenter"style="top:0em;"><spanclass="delimsizingsize1">)</span></span></span><spanclass="msupsub"><spanclass="vlistβˆ’t"><spanclass="vlistβˆ’r"><spanclass="vlist"style="height:0.9653em;"><spanstyle="top:βˆ’3.3639em;marginβˆ’right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingresetβˆ’size6size3mtight"><spanclass="mordmathnormalmtight">c</span></span></span></span></span></span></span></span><spanclass="mspace"style="marginβˆ’right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="marginβˆ’right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:1.2684em;verticalβˆ’align:βˆ’0.3574em;"></span><spanclass="mord"><spanclass="mopennulldelimiter"></span><spanclass="mfrac"><spanclass="vlistβˆ’tvlistβˆ’t2"><spanclass="vlistβˆ’r"><spanclass="vlist"style="height:0.911em;"><spanstyle="top:βˆ’2.6426em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="sizingresetβˆ’size6size3mtight"><spanclass="mordmtight"><spanclass="mordmtight"><spanclass="mordmathnormalmtight">x</span><spanclass="msupsub"><spanclass="vlistβˆ’t"><spanclass="vlistβˆ’r"><spanclass="vlist"style="height:0.782em;"><spanstyle="top:βˆ’2.786em;marginβˆ’right:0.0714em;"><spanclass="pstrut"style="height:2.5em;"></span><spanclass="sizingresetβˆ’size3size1mtight"><spanclass="mordmtight"><spanclass="mordmathnormalmtight">b</span><spanclass="mordmathnormalmtight">c</span></span></span></span></span></span></span></span></span></span></span></span><spanstyle="top:βˆ’3.23em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="fracβˆ’line"style="borderβˆ’bottomβˆ’width:0.04em;"></span></span><spanstyle="top:βˆ’3.394em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="sizingresetβˆ’size6size3mtight"><spanclass="mordmtight"><spanclass="mordmtight"><spanclass="mordmathnormalmtight">x</span><spanclass="msupsub"><spanclass="vlistβˆ’t"><spanclass="vlistβˆ’r"><spanclass="vlist"style="height:0.7385em;"><spanstyle="top:βˆ’2.931em;marginβˆ’right:0.0714em;"><spanclass="pstrut"style="height:2.5em;"></span><spanclass="sizingresetβˆ’size3size1mtight"><spanclass="mordmtight"><spanclass="mordmathnormalmtight">a</span><spanclass="mordmathnormalmtight">c</span></span></span></span></span></span></span></span></span></span></span></span></span><spanclass="vlistβˆ’s">​</span></span><spanclass="vlistβˆ’r"><spanclass="vlist"style="height:0.3574em;"><span></span></span></span></span></span><spanclass="mclosenulldelimiter"></span></span></span></span></span></li><li><spanclass="katex"><spanclass="katexβˆ’mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mrow><mofence="true">(</mo><mfrac><msup><mi>x</mi><mi>a</mi></msup><msup><mi>x</mi><mi>b</mi></msup></mfrac><mofence="true">)</mo></mrow><mo>Γ—</mo><mrow><mofence="true">(</mo><mfrac><msup><mi>x</mi><mi>c</mi></msup><msup><mi>x</mi><mi>d</mi></msup></mfrac><mofence="true">)</mo></mrow><mo>=</mo><mfrac><msup><mi>x</mi><mrow><mi>a</mi><mo>+</mo><mi>c</mi></mrow></msup><msup><mi>x</mi><mrow><mi>b</mi><mo>+</mo><mi>d</mi></mrow></msup></mfrac></mrow><annotationencoding="application/xβˆ’tex">(xaxb)Γ—(xcxd)=xa+cxb+d</annotation></semantics></math></span><spanclass="katexβˆ’html"ariaβˆ’hidden="true"><spanclass="base"><spanclass="strut"style="height:1.2684em;verticalβˆ’align:βˆ’0.3574em;"></span><spanclass="minner"><spanclass="mopendelimcenter"style="top:0em;"><spanclass="delimsizingsize1">(</span></span><spanclass="mord"><spanclass="mopennulldelimiter"></span><spanclass="mfrac"><spanclass="vlistβˆ’tvlistβˆ’t2"><spanclass="vlistβˆ’r"><spanclass="vlist"style="height:0.911em;"><spanstyle="top:βˆ’2.6426em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="sizingresetβˆ’size6size3mtight"><spanclass="mordmtight"><spanclass="mordmtight"><spanclass="mordmathnormalmtight">x</span><spanclass="msupsub"><spanclass="vlistβˆ’t"><spanclass="vlistβˆ’r"><spanclass="vlist"style="height:0.782em;"><spanstyle="top:βˆ’2.786em;marginβˆ’right:0.0714em;"><spanclass="pstrut"style="height:2.5em;"></span><spanclass="sizingresetβˆ’size3size1mtight"><spanclass="mordmathnormalmtight">b</span></span></span></span></span></span></span></span></span></span></span><spanstyle="top:βˆ’3.23em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="fracβˆ’line"style="borderβˆ’bottomβˆ’width:0.04em;"></span></span><spanstyle="top:βˆ’3.394em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="sizingresetβˆ’size6size3mtight"><spanclass="mordmtight"><spanclass="mordmtight"><spanclass="mordmathnormalmtight">x</span><spanclass="msupsub"><spanclass="vlistβˆ’t"><spanclass="vlistβˆ’r"><spanclass="vlist"style="height:0.7385em;"><spanstyle="top:βˆ’2.931em;marginβˆ’right:0.0714em;"><spanclass="pstrut"style="height:2.5em;"></span><spanclass="sizingresetβˆ’size3size1mtight"><spanclass="mordmathnormalmtight">a</span></span></span></span></span></span></span></span></span></span></span></span><spanclass="vlistβˆ’s">​</span></span><spanclass="vlistβˆ’r"><spanclass="vlist"style="height:0.3574em;"><span></span></span></span></span></span><spanclass="mclosenulldelimiter"></span></span><spanclass="mclosedelimcenter"style="top:0em;"><spanclass="delimsizingsize1">)</span></span></span><spanclass="mspace"style="marginβˆ’right:0.2222em;"></span><spanclass="mbin">Γ—</span><spanclass="mspace"style="marginβˆ’right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:1.2684em;verticalβˆ’align:βˆ’0.3574em;"></span><spanclass="minner"><spanclass="mopendelimcenter"style="top:0em;"><spanclass="delimsizingsize1">(</span></span><spanclass="mord"><spanclass="mopennulldelimiter"></span><spanclass="mfrac"><spanclass="vlistβˆ’tvlistβˆ’t2"><spanclass="vlistβˆ’r"><spanclass="vlist"style="height:0.911em;"><spanstyle="top:βˆ’2.6426em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="sizingresetβˆ’size6size3mtight"><spanclass="mordmtight"><spanclass="mordmtight"><spanclass="mordmathnormalmtight">x</span><spanclass="msupsub"><spanclass="vlistβˆ’t"><spanclass="vlistβˆ’r"><spanclass="vlist"style="height:0.782em;"><spanstyle="top:βˆ’2.786em;marginβˆ’right:0.0714em;"><spanclass="pstrut"style="height:2.5em;"></span><spanclass="sizingresetβˆ’size3size1mtight"><spanclass="mordmathnormalmtight">d</span></span></span></span></span></span></span></span></span></span></span><spanstyle="top:βˆ’3.23em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="fracβˆ’line"style="borderβˆ’bottomβˆ’width:0.04em;"></span></span><spanstyle="top:βˆ’3.394em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="sizingresetβˆ’size6size3mtight"><spanclass="mordmtight"><spanclass="mordmtight"><spanclass="mordmathnormalmtight">x</span><spanclass="msupsub"><spanclass="vlistβˆ’t"><spanclass="vlistβˆ’r"><spanclass="vlist"style="height:0.7385em;"><spanstyle="top:βˆ’2.931em;marginβˆ’right:0.0714em;"><spanclass="pstrut"style="height:2.5em;"></span><spanclass="sizingresetβˆ’size3size1mtight"><spanclass="mordmathnormalmtight">c</span></span></span></span></span></span></span></span></span></span></span></span><spanclass="vlistβˆ’s">​</span></span><spanclass="vlistβˆ’r"><spanclass="vlist"style="height:0.3574em;"><span></span></span></span></span></span><spanclass="mclosenulldelimiter"></span></span><spanclass="mclosedelimcenter"style="top:0em;"><spanclass="delimsizingsize1">)</span></span></span><spanclass="mspace"style="marginβˆ’right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="marginβˆ’right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:1.3448em;verticalβˆ’align:βˆ’0.3574em;"></span><spanclass="mord"><spanclass="mopennulldelimiter"></span><spanclass="mfrac"><spanclass="vlistβˆ’tvlistβˆ’t2"><spanclass="vlistβˆ’r"><spanclass="vlist"style="height:0.9874em;"><spanstyle="top:βˆ’2.6426em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="sizingresetβˆ’size6size3mtight"><spanclass="mordmtight"><spanclass="mordmtight"><spanclass="mordmathnormalmtight">x</span><spanclass="msupsub"><spanclass="vlistβˆ’t"><spanclass="vlistβˆ’r"><spanclass="vlist"style="height:0.782em;"><spanstyle="top:βˆ’2.786em;marginβˆ’right:0.0714em;"><spanclass="pstrut"style="height:2.5em;"></span><spanclass="sizingresetβˆ’size3size1mtight"><spanclass="mordmtight"><spanclass="mordmathnormalmtight">b</span><spanclass="mbinmtight">+</span><spanclass="mordmathnormalmtight">d</span></span></span></span></span></span></span></span></span></span></span></span><spanstyle="top:βˆ’3.23em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="fracβˆ’line"style="borderβˆ’bottomβˆ’width:0.04em;"></span></span><spanstyle="top:βˆ’3.394em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="sizingresetβˆ’size6size3mtight"><spanclass="mordmtight"><spanclass="mordmtight"><spanclass="mordmathnormalmtight">x</span><spanclass="msupsub"><spanclass="vlistβˆ’t"><spanclass="vlistβˆ’r"><spanclass="vlist"style="height:0.8477em;"><spanstyle="top:βˆ’2.931em;marginβˆ’right:0.0714em;"><spanclass="pstrut"style="height:2.5em;"></span><spanclass="sizingresetβˆ’size3size1mtight"><spanclass="mordmtight"><spanclass="mordmathnormalmtight">a</span><spanclass="mbinmtight">+</span><spanclass="mordmathnormalmtight">c</span></span></span></span></span></span></span></span></span></span></span></span></span><spanclass="vlistβˆ’s">​</span></span><spanclass="vlistβˆ’r"><spanclass="vlist"style="height:0.3574em;"><span></span></span></span></span></span><spanclass="mclosenulldelimiter"></span></span></span></span></span></li><li><spanclass="katex"><spanclass="katexβˆ’mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mrow><mofence="true">(</mo><mfrac><msup><mi>x</mi><mi>a</mi></msup><msup><mi>x</mi><mi>b</mi></msup></mfrac><mofence="true">)</mo></mrow><mi>c</mi></msup><mo>=</mo><mfrac><msup><mi>x</mi><mrow><mi>a</mi><mi>c</mi></mrow></msup><msup><mi>x</mi><mrow><mi>b</mi><mi>c</mi></mrow></msup></mfrac></mrow><annotationencoding="application/xβˆ’tex">(xaxb)c=xacxbc</annotation></semantics></math></span><spanclass="katexβˆ’html"ariaβˆ’hidden="true"><spanclass="base"><spanclass="strut"style="height:1.3227em;verticalβˆ’align:βˆ’0.3574em;"></span><spanclass="minner"><spanclass="minner"><spanclass="mopendelimcenter"style="top:0em;"><spanclass="delimsizingsize1">(</span></span><spanclass="mord"><spanclass="mopennulldelimiter"></span><spanclass="mfrac"><spanclass="vlistβˆ’tvlistβˆ’t2"><spanclass="vlistβˆ’r"><spanclass="vlist"style="height:0.911em;"><spanstyle="top:βˆ’2.6426em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="sizingresetβˆ’size6size3mtight"><spanclass="mordmtight"><spanclass="mordmtight"><spanclass="mordmathnormalmtight">x</span><spanclass="msupsub"><spanclass="vlistβˆ’t"><spanclass="vlistβˆ’r"><spanclass="vlist"style="height:0.782em;"><spanstyle="top:βˆ’2.786em;marginβˆ’right:0.0714em;"><spanclass="pstrut"style="height:2.5em;"></span><spanclass="sizingresetβˆ’size3size1mtight"><spanclass="mordmathnormalmtight">b</span></span></span></span></span></span></span></span></span></span></span><spanstyle="top:βˆ’3.23em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="fracβˆ’line"style="borderβˆ’bottomβˆ’width:0.04em;"></span></span><spanstyle="top:βˆ’3.394em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="sizingresetβˆ’size6size3mtight"><spanclass="mordmtight"><spanclass="mordmtight"><spanclass="mordmathnormalmtight">x</span><spanclass="msupsub"><spanclass="vlistβˆ’t"><spanclass="vlistβˆ’r"><spanclass="vlist"style="height:0.7385em;"><spanstyle="top:βˆ’2.931em;marginβˆ’right:0.0714em;"><spanclass="pstrut"style="height:2.5em;"></span><spanclass="sizingresetβˆ’size3size1mtight"><spanclass="mordmathnormalmtight">a</span></span></span></span></span></span></span></span></span></span></span></span><spanclass="vlistβˆ’s">​</span></span><spanclass="vlistβˆ’r"><spanclass="vlist"style="height:0.3574em;"><span></span></span></span></span></span><spanclass="mclosenulldelimiter"></span></span><spanclass="mclosedelimcenter"style="top:0em;"><spanclass="delimsizingsize1">)</span></span></span><spanclass="msupsub"><spanclass="vlistβˆ’t"><spanclass="vlistβˆ’r"><spanclass="vlist"style="height:0.9653em;"><spanstyle="top:βˆ’3.3639em;marginβˆ’right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingresetβˆ’size6size3mtight"><spanclass="mordmathnormalmtight">c</span></span></span></span></span></span></span></span><spanclass="mspace"style="marginβˆ’right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="marginβˆ’right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:1.2684em;verticalβˆ’align:βˆ’0.3574em;"></span><spanclass="mord"><spanclass="mopennulldelimiter"></span><spanclass="mfrac"><spanclass="vlistβˆ’tvlistβˆ’t2"><spanclass="vlistβˆ’r"><spanclass="vlist"style="height:0.911em;"><spanstyle="top:βˆ’2.6426em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="sizingresetβˆ’size6size3mtight"><spanclass="mordmtight"><spanclass="mordmtight"><spanclass="mordmathnormalmtight">x</span><spanclass="msupsub"><spanclass="vlistβˆ’t"><spanclass="vlistβˆ’r"><spanclass="vlist"style="height:0.782em;"><spanstyle="top:βˆ’2.786em;marginβˆ’right:0.0714em;"><spanclass="pstrut"style="height:2.5em;"></span><spanclass="sizingresetβˆ’size3size1mtight"><spanclass="mordmtight"><spanclass="mordmathnormalmtight">b</span><spanclass="mordmathnormalmtight">c</span></span></span></span></span></span></span></span></span></span></span></span><spanstyle="top:βˆ’3.23em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="fracβˆ’line"style="borderβˆ’bottomβˆ’width:0.04em;"></span></span><spanstyle="top:βˆ’3.394em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="sizingresetβˆ’size6size3mtight"><spanclass="mordmtight"><spanclass="mordmtight"><spanclass="mordmathnormalmtight">x</span><spanclass="msupsub"><spanclass="vlistβˆ’t"><spanclass="vlistβˆ’r"><spanclass="vlist"style="height:0.7385em;"><spanstyle="top:βˆ’2.931em;marginβˆ’right:0.0714em;"><spanclass="pstrut"style="height:2.5em;"></span><spanclass="sizingresetβˆ’size3size1mtight"><spanclass="mordmtight"><spanclass="mordmathnormalmtight">a</span><spanclass="mordmathnormalmtight">c</span></span></span></span></span></span></span></span></span></span></span></span></span><spanclass="vlistβˆ’s">​</span></span><spanclass="vlistβˆ’r"><spanclass="vlist"style="height:0.3574em;"><span></span></span></span></span></span><spanclass="mclosenulldelimiter"></span></span></span></span></span></li></ul><h3>Q:HowcanIapplythesimplificationtechniquestorealβˆ’worldproblems?</h3><p>A:Youcanapplythesimplificationtechniquestorealβˆ’worldproblemsinvolvingexponentialgrowthanddecay.Forexample,youcanusetheexpressiontomodelthegrowthofapopulationorthedecayofaradioactivesubstance.</p><h2>Conclusion</h2><hr><p>Inthisarticle,weansweredsomefrequentlyaskedquestionsrelatedtothesimplificationofacomplexalgebraicexpressioninvolvingexponentsandfractions.Weprovidedastepβˆ’byβˆ’stepexplanationofthesimplificationprocessandofferedexamplesofsimplifyingexpressionsinvolvingexponentsandfractions.Wealsodiscussedthesignificanceoftheexpressionandprovidedtipsonhowtoapplythesimplificationtechniquestorealβˆ’worldproblems.</p><h2>FinalAnswer</h2><hr><p>Thefinalsimplifiedexpressionis:</p><pclass=β€²katexβˆ’blockβ€²><spanclass="katexβˆ’display"><spanclass="katex"><spanclass="katexβˆ’mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><msup><mi>x</mi><mrow><msup><mi>c</mi><mn>3</mn></msup><mo>βˆ’</mo><msup><mi>a</mi><mn>2</mn></msup><msup><mi>c</mi><mn>2</mn></msup><mo>βˆ’</mo><msup><mi>b</mi><mn>3</mn></msup></mrow></msup></mrow><annotationencoding="application/xβˆ’tex">xc3βˆ’a2c2βˆ’b3</annotation></semantics></math></span><spanclass="katexβˆ’html"ariaβˆ’hidden="true"><spanclass="base"><spanclass="strut"style="height:1.0369em;"></span><spanclass="mord"><spanclass="mordmathnormal">x</span><spanclass="msupsub"><spanclass="vlistβˆ’t"><spanclass="vlistβˆ’r"><spanclass="vlist"style="height:1.0369em;"><spanstyle="top:βˆ’3.113em;marginβˆ’right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingresetβˆ’size6size3mtight"><spanclass="mordmtight"><spanclass="mordmtight"><spanclass="mordmathnormalmtight">c</span><spanclass="msupsub"><spanclass="vlistβˆ’t"><spanclass="vlistβˆ’r"><spanclass="vlist"style="height:0.8913em;"><spanstyle="top:βˆ’2.931em;marginβˆ’right:0.0714em;"><spanclass="pstrut"style="height:2.5em;"></span><spanclass="sizingresetβˆ’size3size1mtight"><spanclass="mordmtight">3</span></span></span></span></span></span></span></span><spanclass="mbinmtight">βˆ’</span><spanclass="mordmtight"><spanclass="mordmathnormalmtight">a</span><spanclass="msupsub"><spanclass="vlistβˆ’t"><spanclass="vlistβˆ’r"><spanclass="vlist"style="height:0.8913em;"><spanstyle="top:βˆ’2.931em;marginβˆ’right:0.0714em;"><spanclass="pstrut"style="height:2.5em;"></span><spanclass="sizingresetβˆ’size3size1mtight"><spanclass="mordmtight">2</span></span></span></span></span></span></span></span><spanclass="mordmtight"><spanclass="mordmathnormalmtight">c</span><spanclass="msupsub"><spanclass="vlistβˆ’t"><spanclass="vlistβˆ’r"><spanclass="vlist"style="height:0.8913em;"><spanstyle="top:βˆ’2.931em;marginβˆ’right:0.0714em;"><spanclass="pstrut"style="height:2.5em;"></span><spanclass="sizingresetβˆ’size3size1mtight"><spanclass="mordmtight">2</span></span></span></span></span></span></span></span><spanclass="mbinmtight">βˆ’</span><spanclass="mordmtight"><spanclass="mordmathnormalmtight">b</span><spanclass="msupsub"><spanclass="vlistβˆ’t"><spanclass="vlistβˆ’r"><spanclass="vlist"style="height:0.8913em;"><spanstyle="top:βˆ’2.931em;marginβˆ’right:0.0714em;"><spanclass="pstrut"style="height:2.5em;"></span><spanclass="sizingresetβˆ’size3size1mtight"><spanclass="mordmtight">3</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></p><p>Thisexpressioncanbeusedtomodelrealβˆ’worldproblemsinvolvingexponentialgrowthanddecay.</p>x^{c^3 - a^2c^2 - b^3} </span></p> <h3>Q: How did you simplify the expression?</h3> <p>A: We used the properties of exponents to simplify each term and then combined them. We also canceled out common factors to simplify the expression further.</p> <h3>Q: What are the properties of exponents that you used?</h3> <p>A: We used the following properties of exponents:</p> <ul> <li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>x</mi><mi>a</mi></msup><msup><mo stretchy="false">)</mo><mi>b</mi></msup><mo>=</mo><msup><mi>x</mi><mrow><mi>a</mi><mi>b</mi></mrow></msup></mrow><annotation encoding="application/x-tex">(x^a)^b = x^{ab}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0991em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">a</span></span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">b</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8491em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">ab</span></span></span></span></span></span></span></span></span></span></span></span></li> <li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><msup><mi>x</mi><mi>a</mi></msup><msup><mi>x</mi><mi>b</mi></msup></mfrac><mo>=</mo><msup><mi>x</mi><mrow><mi>a</mi><mo>βˆ’</mo><mi>b</mi></mrow></msup></mrow><annotation encoding="application/x-tex">\frac{x^a}{x^b} = x^{a-b}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2684em;vertical-align:-0.3574em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.911em;"><span style="top:-2.6426em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.782em;"><span style="top:-2.786em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight">b</span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7385em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight">a</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3574em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8491em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">a</span><span class="mbin mtight">βˆ’</span><span class="mord mathnormal mtight">b</span></span></span></span></span></span></span></span></span></span></span></span></li> </ul> <h3>Q: Can you explain the simplification process in more detail?</h3> <p>A: Here is a step-by-step explanation of the simplification process:</p> <ol> <li>Simplify each term using the properties of exponents.</li> <li>Combine the simplified terms.</li> <li>Cancel out common factors.</li> <li>Simplify the expression further.</li> </ol> <h3>Q: What is the significance of the expression?</h3> <p>A: The expression is a product of three terms, each containing exponents and fractions. The expression can be used to model real-world problems involving exponential growth and decay.</p> <h3>Q: Can you provide more examples of simplifying expressions involving exponents and fractions?</h3> <p>A: Yes, here are a few examples:</p> <ul> <li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mrow><mo fence="true">(</mo><mfrac><msup><mi>x</mi><mi>a</mi></msup><msup><mi>x</mi><mi>b</mi></msup></mfrac><mo fence="true">)</mo></mrow><mi>c</mi></msup><mo>=</mo><mfrac><msup><mi>x</mi><mrow><mi>a</mi><mi>c</mi></mrow></msup><msup><mi>x</mi><mrow><mi>b</mi><mi>c</mi></mrow></msup></mfrac></mrow><annotation encoding="application/x-tex">\left(\frac{x^a}{x^b}\right)^c = \frac{x^{ac}}{x^{bc}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.3227em;vertical-align:-0.3574em;"></span><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.911em;"><span style="top:-2.6426em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.782em;"><span style="top:-2.786em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight">b</span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7385em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight">a</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3574em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">)</span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9653em;"><span style="top:-3.3639em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">c</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.2684em;vertical-align:-0.3574em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.911em;"><span style="top:-2.6426em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.782em;"><span style="top:-2.786em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">b</span><span class="mord mathnormal mtight">c</span></span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7385em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">a</span><span class="mord mathnormal mtight">c</span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3574em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></li> <li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mrow><mo fence="true">(</mo><mfrac><msup><mi>x</mi><mi>a</mi></msup><msup><mi>x</mi><mi>b</mi></msup></mfrac><mo fence="true">)</mo></mrow><mo>Γ—</mo><mrow><mo fence="true">(</mo><mfrac><msup><mi>x</mi><mi>c</mi></msup><msup><mi>x</mi><mi>d</mi></msup></mfrac><mo fence="true">)</mo></mrow><mo>=</mo><mfrac><msup><mi>x</mi><mrow><mi>a</mi><mo>+</mo><mi>c</mi></mrow></msup><msup><mi>x</mi><mrow><mi>b</mi><mo>+</mo><mi>d</mi></mrow></msup></mfrac></mrow><annotation encoding="application/x-tex">\left(\frac{x^a}{x^b}\right) \times \left(\frac{x^c}{x^d}\right) = \frac{x^{a+c}}{x^{b+d}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2684em;vertical-align:-0.3574em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.911em;"><span style="top:-2.6426em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.782em;"><span style="top:-2.786em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight">b</span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7385em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight">a</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3574em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">)</span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">Γ—</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.2684em;vertical-align:-0.3574em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.911em;"><span style="top:-2.6426em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.782em;"><span style="top:-2.786em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight">d</span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7385em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight">c</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3574em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">)</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.3448em;vertical-align:-0.3574em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9874em;"><span style="top:-2.6426em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.782em;"><span style="top:-2.786em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">b</span><span class="mbin mtight">+</span><span class="mord mathnormal mtight">d</span></span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8477em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">a</span><span class="mbin mtight">+</span><span class="mord mathnormal mtight">c</span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3574em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></li> <li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mrow><mo fence="true">(</mo><mfrac><msup><mi>x</mi><mi>a</mi></msup><msup><mi>x</mi><mi>b</mi></msup></mfrac><mo fence="true">)</mo></mrow><mi>c</mi></msup><mo>=</mo><mfrac><msup><mi>x</mi><mrow><mi>a</mi><mi>c</mi></mrow></msup><msup><mi>x</mi><mrow><mi>b</mi><mi>c</mi></mrow></msup></mfrac></mrow><annotation encoding="application/x-tex">\left(\frac{x^a}{x^b}\right)^c = \frac{x^{ac}}{x^{bc}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.3227em;vertical-align:-0.3574em;"></span><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.911em;"><span style="top:-2.6426em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.782em;"><span style="top:-2.786em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight">b</span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7385em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight">a</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3574em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">)</span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9653em;"><span style="top:-3.3639em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">c</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.2684em;vertical-align:-0.3574em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.911em;"><span style="top:-2.6426em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.782em;"><span style="top:-2.786em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">b</span><span class="mord mathnormal mtight">c</span></span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7385em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">a</span><span class="mord mathnormal mtight">c</span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3574em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></li> </ul> <h3>Q: How can I apply the simplification techniques to real-world problems?</h3> <p>A: You can apply the simplification techniques to real-world problems involving exponential growth and decay. For example, you can use the expression to model the growth of a population or the decay of a radioactive substance.</p> <h2>Conclusion</h2> <hr> <p>In this article, we answered some frequently asked questions related to the simplification of a complex algebraic expression involving exponents and fractions. We provided a step-by-step explanation of the simplification process and offered examples of simplifying expressions involving exponents and fractions. We also discussed the significance of the expression and provided tips on how to apply the simplification techniques to real-world problems.</p> <h2>Final Answer</h2> <hr> <p>The final simplified expression is:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msup><mi>x</mi><mrow><msup><mi>c</mi><mn>3</mn></msup><mo>βˆ’</mo><msup><mi>a</mi><mn>2</mn></msup><msup><mi>c</mi><mn>2</mn></msup><mo>βˆ’</mo><msup><mi>b</mi><mn>3</mn></msup></mrow></msup></mrow><annotation encoding="application/x-tex">x^{c^3 - a^2c^2 - b^3} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0369em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.0369em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">c</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span><span class="mbin mtight">βˆ’</span><span class="mord mtight"><span class="mord mathnormal mtight">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord mtight"><span class="mord mathnormal mtight">c</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mbin mtight">βˆ’</span><span class="mord mtight"><span class="mord mathnormal mtight">b</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></p> <p>This expression can be used to model real-world problems involving exponential growth and decay.</p>