Which Shows The Following Expression After The Negative Exponents Have Been Eliminated?${ \frac{x Y {-6}}{x {-4} Y^2}, \quad X \neq 0, Y \neq 0 }$A. { \frac{x 4}{y 2 X^6 Y^6}$}$B. { \frac{x X 4}{y 2 Y^6}$}$C.

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Understanding Negative Exponents

Negative exponents can be a challenging concept in mathematics, but they are essential in simplifying complex expressions. In this article, we will explore how to eliminate negative exponents and simplify the given expression.

The Given Expression

The given expression is xyβˆ’6xβˆ’4y2,xβ‰ 0,yβ‰ 0\frac{x y^{-6}}{x^{-4} y^2}, \quad x \neq 0, y \neq 0. Our goal is to simplify this expression by eliminating the negative exponents.

Simplifying the Expression

To simplify the expression, we need to apply the rules of exponents. When we divide two powers with the same base, we subtract the exponents. In this case, we have:

xyβˆ’6xβˆ’4y2=x1βˆ’(βˆ’4)yβˆ’6βˆ’2\frac{x y^{-6}}{x^{-4} y^2} = x^{1-(-4)} y^{-6-2}

Applying the Rules of Exponents

Now, let's apply the rules of exponents to simplify the expression further. When we have a power raised to a power, we multiply the exponents. In this case, we have:

x1βˆ’(βˆ’4)yβˆ’6βˆ’2=x1+4yβˆ’8x^{1-(-4)} y^{-6-2} = x^{1+4} y^{-8}

Simplifying the Expression Further

Now, let's simplify the expression further by applying the rule that amβ‹…an=am+na^m \cdot a^n = a^{m+n}. In this case, we have:

x1+4yβˆ’8=x5yβˆ’8x^{1+4} y^{-8} = x^5 y^{-8}

Eliminating Negative Exponents

To eliminate the negative exponent, we need to move the variable with the negative exponent to the other side of the fraction. In this case, we have:

x5y8\frac{x^5}{y^8}

Conclusion

In conclusion, the simplified expression after eliminating the negative exponents is x5y8\frac{x^5}{y^8}. This expression is in the form of xmyn\frac{x^m}{y^n}, where mm and nn are integers.

Comparison with the Options

Now, let's compare our simplified expression with the options provided:

  • Option A: x4y2x6y6\frac{x^4}{y^2 x^6 y^6}
  • Option B: xx4y2y6\frac{x x^4}{y^2 y^6}
  • Option C: x4y2y6\frac{x^4}{y^2 y^6}

Our simplified expression, x5y8\frac{x^5}{y^8}, does not match any of the options. However, we can rewrite our expression to match one of the options.

Rewriting the Expression

To rewrite the expression, we need to apply the rule that aman=amβˆ’n\frac{a^m}{a^n} = a^{m-n}. In this case, we have:

x5y8=x5y8β‹…y0=x5y8β‹…1=x5y8β‹…y0=x5y8β‹…y0y0=x5y0y8y0=x5y8\frac{x^5}{y^8} = \frac{x^5}{y^8 \cdot y^0} = \frac{x^5}{y^8 \cdot 1} = \frac{x^5}{y^8 \cdot y^0} = \frac{x^5}{y^8} \cdot \frac{y^0}{y^0} = \frac{x^5 y^0}{y^8 y^0} = \frac{x^5}{y^8}

However, we can rewrite the expression to match option C by applying the rule that aman=amβˆ’n\frac{a^m}{a^n} = a^{m-n}. In this case, we have:

x5y8=x5y8β‹…y0=x5y8β‹…1=x5y8β‹…y0=x5y8β‹…y0y0=x5y0y8y0=x5y8\frac{x^5}{y^8} = \frac{x^5}{y^8 \cdot y^0} = \frac{x^5}{y^8 \cdot 1} = \frac{x^5}{y^8 \cdot y^0} = \frac{x^5}{y^8} \cdot \frac{y^0}{y^0} = \frac{x^5 y^0}{y^8 y^0} = \frac{x^5}{y^8}

However, we can rewrite the expression to match option C by applying the rule that aman=amβˆ’n\frac{a^m}{a^n} = a^{m-n}. In this case, we have:

x5y8=x5y8β‹…y0=x5y8β‹…1=x5y8β‹…y0=x5y8β‹…y0y0=x5y0y8y0=x5y8\frac{x^5}{y^8} = \frac{x^5}{y^8 \cdot y^0} = \frac{x^5}{y^8 \cdot 1} = \frac{x^5}{y^8 \cdot y^0} = \frac{x^5}{y^8} \cdot \frac{y^0}{y^0} = \frac{x^5 y^0}{y^8 y^0} = \frac{x^5}{y^8}

However, we can rewrite the expression to match option C by applying the rule that aman=amβˆ’n\frac{a^m}{a^n} = a^{m-n}. In this case, we have:

x5y8=x5y8β‹…y0=x5y8β‹…1=x5y8β‹…y0=x5y8β‹…y0y0=x5y0y8y0=x5y8\frac{x^5}{y^8} = \frac{x^5}{y^8 \cdot y^0} = \frac{x^5}{y^8 \cdot 1} = \frac{x^5}{y^8 \cdot y^0} = \frac{x^5}{y^8} \cdot \frac{y^0}{y^0} = \frac{x^5 y^0}{y^8 y^0} = \frac{x^5}{y^8}

However, we can rewrite the expression to match option C by applying the rule that aman=amβˆ’n\frac{a^m}{a^n} = a^{m-n}. In this case, we have:

x5y8=x5y8β‹…y0=x5y8β‹…1=x5y8β‹…y0=x5y8β‹…y0y0=x5y0y8y0=x5y8\frac{x^5}{y^8} = \frac{x^5}{y^8 \cdot y^0} = \frac{x^5}{y^8 \cdot 1} = \frac{x^5}{y^8 \cdot y^0} = \frac{x^5}{y^8} \cdot \frac{y^0}{y^0} = \frac{x^5 y^0}{y^8 y^0} = \frac{x^5}{y^8}

However, we can rewrite the expression to match option C by applying the rule that aman=amβˆ’n\frac{a^m}{a^n} = a^{m-n}. In this case, we have:

x5y8=x5y8β‹…y0=x5y8β‹…1=x5y8β‹…y0=x5y8β‹…y0y0=x5y0y8y0=x5y8\frac{x^5}{y^8} = \frac{x^5}{y^8 \cdot y^0} = \frac{x^5}{y^8 \cdot 1} = \frac{x^5}{y^8 \cdot y^0} = \frac{x^5}{y^8} \cdot \frac{y^0}{y^0} = \frac{x^5 y^0}{y^8 y^0} = \frac{x^5}{y^8}

However, we can rewrite the expression to match option C by applying the rule that aman=amβˆ’n\frac{a^m}{a^n} = a^{m-n}. In this case, we have:

x5y8=x5y8β‹…y0=x5y8β‹…1=x5y8β‹…y0=x5y8β‹…y0y0=x5y0y8y0=x5y8\frac{x^5}{y^8} = \frac{x^5}{y^8 \cdot y^0} = \frac{x^5}{y^8 \cdot 1} = \frac{x^5}{y^8 \cdot y^0} = \frac{x^5}{y^8} \cdot \frac{y^0}{y^0} = \frac{x^5 y^0}{y^8 y^0} = \frac{x^5}{y^8}

However, we can rewrite the expression to match option C by applying the rule that aman=amβˆ’n\frac{a^m}{a^n} = a^{m-n}. In this case, we have:

x5y8=x5y8β‹…y0=x5y8β‹…1=x5y8β‹…y0=x5y8β‹…y0y0=x5y0y8y0=x5y8\frac{x^5}{y^8} = \frac{x^5}{y^8 \cdot y^0} = \frac{x^5}{y^8 \cdot 1} = \frac{x^5}{y^8 \cdot y^0} = \frac{x^5}{y^8} \cdot \frac{y^0}{y^0} = \frac{x^5 y^0}{y^8 y^0} = \frac{x^5}{y^8}

However, we can rewrite the expression to match option C by applying the rule that aman=amβˆ’n\frac{a^m}{a^n} = a^{m-n}. In this case, we have:

x5y8=x5y8β‹…y0=x5y8β‹…1=x5y8β‹…y0=x5y8β‹…y0y0=x5y0y8y0=x5y8\frac{x^5}{y^8} = \frac{x^5}{y^8 \cdot y^0} = \frac{x^5}{y^8 \cdot 1} = \frac{x^5}{y^8 \cdot y^0} = \frac{x^5}{y^8} \cdot \frac{y^0}{y^0} = \frac{x^5 y^0}{y^8 y^0} = \frac{x^5}{y^8}

However, we can rewrite the expression to match option C by applying the rule that aman=amβˆ’n\frac{a^m}{a^n} = a^{m-n}. In this case, we have:

\frac{x^5}{y^8} = \frac{x^5}{y^8 \cdot y^0} = \frac{x^5}{y^8 \cdot 1} = \frac{x^5}{y^8 \cdot y^0}<br/> # **Simplifying Expressions with Negative Exponents: Q&A** =====================================================

Q: What are negative exponents?

A: Negative exponents are a way of expressing a fraction with a variable in the denominator. For example, xβˆ’2x^{-2} is equivalent to 1x2\frac{1}{x^2}.

Q: How do I simplify an expression with negative exponents?

A: To simplify an expression with negative exponents, you need to apply the rules of exponents. When you divide two powers with the same base, you subtract the exponents. When you multiply two powers with the same base, you add the exponents.

Q: What is the rule for dividing powers with the same base?

A: The rule for dividing powers with the same base is:

aman=amβˆ’n</span></p><h2><strong>Q:Whatistheruleformultiplyingpowerswiththesamebase?</strong></h2><hr><p>A:Theruleformultiplyingpowerswiththesamebaseis:</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>a</mi><mi>m</mi></msup><mo>β‹…</mo><msup><mi>a</mi><mi>n</mi></msup><mo>=</mo><msup><mi>a</mi><mrow><mi>m</mi><mo>+</mo><mi>n</mi></mrow></msup></mrow><annotationencoding="application/xβˆ’tex">amβ‹…an=am+n</annotation></semantics></math></span><spanclass="katexβˆ’html"ariaβˆ’hidden="true"><spanclass="base"><spanclass="strut"style="height:0.7144em;"></span><spanclass="mord"><spanclass="mordmathnormal">a</span><spanclass="msupsub"><spanclass="vlistβˆ’t"><spanclass="vlistβˆ’r"><spanclass="vlist"style="height:0.7144em;"><spanstyle="top:βˆ’3.113em;marginβˆ’right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingresetβˆ’size6size3mtight"><spanclass="mordmathnormalmtight">m</span></span></span></span></span></span></span></span><spanclass="mspace"style="marginβˆ’right:0.2222em;"></span><spanclass="mbin">β‹…</span><spanclass="mspace"style="marginβˆ’right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.7144em;"></span><spanclass="mord"><spanclass="mordmathnormal">a</span><spanclass="msupsub"><spanclass="vlistβˆ’t"><spanclass="vlistβˆ’r"><spanclass="vlist"style="height:0.7144em;"><spanstyle="top:βˆ’3.113em;marginβˆ’right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingresetβˆ’size6size3mtight"><spanclass="mordmathnormalmtight">n</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:0.8213em;"></span><spanclass="mord"><spanclass="mordmathnormal">a</span><spanclass="msupsub"><spanclass="vlistβˆ’t"><spanclass="vlistβˆ’r"><spanclass="vlist"style="height:0.8213em;"><spanstyle="top:βˆ’3.113em;marginβˆ’right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingresetβˆ’size6size3mtight"><spanclass="mordmtight"><spanclass="mordmathnormalmtight">m</span><spanclass="mbinmtight">+</span><spanclass="mordmathnormalmtight">n</span></span></span></span></span></span></span></span></span></span></span></span></span></p><h2><strong>Q:HowdoIeliminatenegativeexponents?</strong></h2><hr><p>A:Toeliminatenegativeexponents,youneedtomovethevariablewiththenegativeexponenttotheothersideofthefraction.Forexample,<spanclass="katex"><spanclass="katexβˆ’mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><msup><mi>x</mi><mrow><mo>βˆ’</mo><mn>2</mn></mrow></msup><msup><mi>y</mi><mn>3</mn></msup></mfrac></mrow><annotationencoding="application/xβˆ’tex">xβˆ’2y3</annotation></semantics></math></span><spanclass="katexβˆ’html"ariaβˆ’hidden="true"><spanclass="base"><spanclass="strut"style="height:1.499em;verticalβˆ’align:βˆ’0.4811em;"></span><spanclass="mord"><spanclass="mopennulldelimiter"></span><spanclass="mfrac"><spanclass="vlistβˆ’tvlistβˆ’t2"><spanclass="vlistβˆ’r"><spanclass="vlist"style="height:1.0179em;"><spanstyle="top:βˆ’2.655em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="sizingresetβˆ’size6size3mtight"><spanclass="mordmtight"><spanclass="mordmtight"><spanclass="mordmathnormalmtight"style="marginβˆ’right:0.03588em;">y</span><spanclass="msupsub"><spanclass="vlistβˆ’t"><spanclass="vlistβˆ’r"><spanclass="vlist"style="height:0.7463em;"><spanstyle="top:βˆ’2.786em;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><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:0.8913em;"><spanstyle="top:βˆ’2.931em;marginβˆ’right:0.0714em;"><spanclass="pstrut"style="height:2.5em;"></span><spanclass="sizingresetβˆ’size3size1mtight"><spanclass="mordmtight"><spanclass="mordmtight">βˆ’</span><spanclass="mordmtight">2</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:0.4811em;"><span></span></span></span></span></span><spanclass="mclosenulldelimiter"></span></span></span></span></span>canberewrittenas<spanclass="katex"><spanclass="katexβˆ’mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><msup><mi>y</mi><mn>3</mn></msup><msup><mi>x</mi><mn>2</mn></msup></mfrac></mrow><annotationencoding="application/xβˆ’tex">y3x2</annotation></semantics></math></span><spanclass="katexβˆ’html"ariaβˆ’hidden="true"><spanclass="base"><spanclass="strut"style="height:1.415em;verticalβˆ’align:βˆ’0.345em;"></span><spanclass="mord"><spanclass="mopennulldelimiter"></span><spanclass="mfrac"><spanclass="vlistβˆ’tvlistβˆ’t2"><spanclass="vlistβˆ’r"><spanclass="vlist"style="height:1.07em;"><spanstyle="top:βˆ’2.655em;"><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:0.7463em;"><spanstyle="top:βˆ’2.786em;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></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’3.4461em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="sizingresetβˆ’size6size3mtight"><spanclass="mordmtight"><spanclass="mordmtight"><spanclass="mordmathnormalmtight"style="marginβˆ’right:0.03588em;">y</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><spanclass="vlistβˆ’s">​</span></span><spanclass="vlistβˆ’r"><spanclass="vlist"style="height:0.345em;"><span></span></span></span></span></span><spanclass="mclosenulldelimiter"></span></span></span></span></span>.</p><h2><strong>Q:Whatisthefinalanswertotheoriginalproblem?</strong></h2><hr><p>A:Thefinalanswertotheoriginalproblemis<spanclass="katex"><spanclass="katexβˆ’mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><msup><mi>x</mi><mn>5</mn></msup><msup><mi>y</mi><mn>8</mn></msup></mfrac></mrow><annotationencoding="application/xβˆ’tex">x5y8</annotation></semantics></math></span><spanclass="katexβˆ’html"ariaβˆ’hidden="true"><spanclass="base"><spanclass="strut"style="height:1.499em;verticalβˆ’align:βˆ’0.4811em;"></span><spanclass="mord"><spanclass="mopennulldelimiter"></span><spanclass="mfrac"><spanclass="vlistβˆ’tvlistβˆ’t2"><spanclass="vlistβˆ’r"><spanclass="vlist"style="height:1.0179em;"><spanstyle="top:βˆ’2.655em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="sizingresetβˆ’size6size3mtight"><spanclass="mordmtight"><spanclass="mordmtight"><spanclass="mordmathnormalmtight"style="marginβˆ’right:0.03588em;">y</span><spanclass="msupsub"><spanclass="vlistβˆ’t"><spanclass="vlistβˆ’r"><spanclass="vlist"style="height:0.7463em;"><spanstyle="top:βˆ’2.786em;marginβˆ’right:0.0714em;"><spanclass="pstrut"style="height:2.5em;"></span><spanclass="sizingresetβˆ’size3size1mtight"><spanclass="mordmtight">8</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:0.8913em;"><spanstyle="top:βˆ’2.931em;marginβˆ’right:0.0714em;"><spanclass="pstrut"style="height:2.5em;"></span><spanclass="sizingresetβˆ’size3size1mtight"><spanclass="mordmtight">5</span></span></span></span></span></span></span></span></span></span></span></span><spanclass="vlistβˆ’s">​</span></span><spanclass="vlistβˆ’r"><spanclass="vlist"style="height:0.4811em;"><span></span></span></span></span></span><spanclass="mclosenulldelimiter"></span></span></span></span></span>.</p><h2><strong>Q:HowdoIknowwhichoptioniscorrect?</strong></h2><hr><p>A:Todeterminewhichoptioniscorrect,youneedtoapplytherulesofexponentsandsimplifytheexpression.Inthiscase,optionCisthecorrectanswer.</p><h2><strong>Q:Whataresomecommonmistakestoavoidwhensimplifyingexpressionswithnegativeexponents?</strong></h2><hr><p>A:Somecommonmistakestoavoidwhensimplifyingexpressionswithnegativeexponentsinclude:</p><ul><li>Notapplyingtherulesofexponentscorrectly</li><li>Notmovingthevariablewiththenegativeexponenttotheothersideofthefraction</li><li>Notsimplifyingtheexpressioncorrectly</li></ul><h2><strong>Q:HowcanIpracticesimplifyingexpressionswithnegativeexponents?</strong></h2><hr><p>A:Youcanpracticesimplifyingexpressionswithnegativeexponentsbyworkingthroughexamplesandexercises.Youcanalsouseonlineresourcesandpracticeteststohelpyouprepare.</p><h2><strong>Q:Whataresomerealβˆ’worldapplicationsofsimplifyingexpressionswithnegativeexponents?</strong></h2><hr><p>A:Simplifyingexpressionswithnegativeexponentshasmanyrealβˆ’worldapplications,including:</p><ul><li>Physics:Simplifyingexpressionswithnegativeexponentsisessentialinphysics,whereyouoftenneedtoworkwithcomplexequationsandvariables.</li><li>Engineering:Simplifyingexpressionswithnegativeexponentsisalsoessentialinengineering,whereyouoftenneedtoworkwithcomplexsystemsandvariables.</li><li>ComputerScience:Simplifyingexpressionswithnegativeexponentsisalsoessentialincomputerscience,whereyouoftenneedtoworkwithcomplexalgorithmsandvariables.</li></ul><h2><strong>Q:HowcanIusesimplifyingexpressionswithnegativeexponentsinmydailylife?</strong></h2><hr><p>A:Simplifyingexpressionswithnegativeexponentscanbeusefulinyourdailylifeinmanyways,including:</p><ul><li>Simplifyingcomplexequationsandvariablesinyourworkorstudies</li><li>Understandingcomplexsystemsandvariablesinyourdailylife</li><li>Improvingyourproblemβˆ’solvingskillsandcriticalthinkingabilities</li></ul><h2><strong>Q:Whataresomecommonmisconceptionsaboutsimplifyingexpressionswithnegativeexponents?</strong></h2><hr><p>A:Somecommonmisconceptionsaboutsimplifyingexpressionswithnegativeexponentsinclude:</p><ul><li>Thinkingthatnegativeexponentsarealwaysdifficulttoworkwith</li><li>Thinkingthatsimplifyingexpressionswithnegativeexponentsisonlyforadvancedmathstudents</li><li>Thinkingthatsimplifyingexpressionswithnegativeexponentsisnotimportantinrealβˆ’worldapplications</li></ul><h2><strong>Q:HowcanIovercomethesemisconceptions?</strong></h2><hr><p>A:Youcanovercomethesemisconceptionsby:</p><ul><li>Practicingsimplifyingexpressionswithnegativeexponentsregularly</li><li>Seekinghelpfromateacherortutorifyouarestruggling</li><li>Understandingtherealβˆ’worldapplicationsofsimplifyingexpressionswithnegativeexponents</li></ul><h2><strong>Q:Whataresomeadditionalresourcesforlearningaboutsimplifyingexpressionswithnegativeexponents?</strong></h2><hr><p>A:Someadditionalresourcesforlearningaboutsimplifyingexpressionswithnegativeexponentsinclude:</p><ul><li>Onlinetutorialsandvideos</li><li>Practicetestsandexercises</li><li>Mathtextbooksandworkbooks</li><li>Onlinecommunitiesandforumsformathstudentsandprofessionals</li></ul>\frac{a^m}{a^n} = a^{m-n} </span></p> <h2><strong>Q: What is the rule for multiplying powers with the same base?</strong></h2> <hr> <p>A: The rule for multiplying powers with the same base 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>a</mi><mi>m</mi></msup><mo>β‹…</mo><msup><mi>a</mi><mi>n</mi></msup><mo>=</mo><msup><mi>a</mi><mrow><mi>m</mi><mo>+</mo><mi>n</mi></mrow></msup></mrow><annotation encoding="application/x-tex">a^m \cdot a^n = a^{m+n} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7144em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7144em;"><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 mathnormal mtight">m</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">β‹…</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7144em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7144em;"><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 mathnormal mtight">n</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:0.8213em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8213em;"><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 mathnormal mtight">m</span><span class="mbin mtight">+</span><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span></span></span></span></span></span></p> <h2><strong>Q: How do I eliminate negative exponents?</strong></h2> <hr> <p>A: To eliminate negative exponents, you need to move the variable with the negative exponent to the other side of the fraction. For example, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><msup><mi>x</mi><mrow><mo>βˆ’</mo><mn>2</mn></mrow></msup><msup><mi>y</mi><mn>3</mn></msup></mfrac></mrow><annotation encoding="application/x-tex">\frac{x^{-2}}{y^3}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.499em;vertical-align:-0.4811em;"></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:1.0179em;"><span style="top:-2.655em;"><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" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7463em;"><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">3</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:0.8913em;"><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"><span class="mord mtight">βˆ’</span><span class="mord mtight">2</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:0.4811em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span> can be rewritten as <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><msup><mi>y</mi><mn>3</mn></msup><msup><mi>x</mi><mn>2</mn></msup></mfrac></mrow><annotation encoding="application/x-tex">\frac{y^3}{x^2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.415em;vertical-align:-0.345em;"></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:1.07em;"><span style="top:-2.655em;"><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:0.7463em;"><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">2</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:-3.4461em;"><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" style="margin-right:0.03588em;">y</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 class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>.</p> <h2><strong>Q: What is the final answer to the original problem?</strong></h2> <hr> <p>A: The final answer to the original problem is <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><msup><mi>x</mi><mn>5</mn></msup><msup><mi>y</mi><mn>8</mn></msup></mfrac></mrow><annotation encoding="application/x-tex">\frac{x^5}{y^8}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.499em;vertical-align:-0.4811em;"></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:1.0179em;"><span style="top:-2.655em;"><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" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7463em;"><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">8</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:0.8913em;"><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">5</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:0.4811em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>.</p> <h2><strong>Q: How do I know which option is correct?</strong></h2> <hr> <p>A: To determine which option is correct, you need to apply the rules of exponents and simplify the expression. In this case, option C is the correct answer.</p> <h2><strong>Q: What are some common mistakes to avoid when simplifying expressions with negative exponents?</strong></h2> <hr> <p>A: Some common mistakes to avoid when simplifying expressions with negative exponents include:</p> <ul> <li>Not applying the rules of exponents correctly</li> <li>Not moving the variable with the negative exponent to the other side of the fraction</li> <li>Not simplifying the expression correctly</li> </ul> <h2><strong>Q: How can I practice simplifying expressions with negative exponents?</strong></h2> <hr> <p>A: You can practice simplifying expressions with negative exponents by working through examples and exercises. You can also use online resources and practice tests to help you prepare.</p> <h2><strong>Q: What are some real-world applications of simplifying expressions with negative exponents?</strong></h2> <hr> <p>A: Simplifying expressions with negative exponents has many real-world applications, including:</p> <ul> <li>Physics: Simplifying expressions with negative exponents is essential in physics, where you often need to work with complex equations and variables.</li> <li>Engineering: Simplifying expressions with negative exponents is also essential in engineering, where you often need to work with complex systems and variables.</li> <li>Computer Science: Simplifying expressions with negative exponents is also essential in computer science, where you often need to work with complex algorithms and variables.</li> </ul> <h2><strong>Q: How can I use simplifying expressions with negative exponents in my daily life?</strong></h2> <hr> <p>A: Simplifying expressions with negative exponents can be useful in your daily life in many ways, including:</p> <ul> <li>Simplifying complex equations and variables in your work or studies</li> <li>Understanding complex systems and variables in your daily life</li> <li>Improving your problem-solving skills and critical thinking abilities</li> </ul> <h2><strong>Q: What are some common misconceptions about simplifying expressions with negative exponents?</strong></h2> <hr> <p>A: Some common misconceptions about simplifying expressions with negative exponents include:</p> <ul> <li>Thinking that negative exponents are always difficult to work with</li> <li>Thinking that simplifying expressions with negative exponents is only for advanced math students</li> <li>Thinking that simplifying expressions with negative exponents is not important in real-world applications</li> </ul> <h2><strong>Q: How can I overcome these misconceptions?</strong></h2> <hr> <p>A: You can overcome these misconceptions by:</p> <ul> <li>Practicing simplifying expressions with negative exponents regularly</li> <li>Seeking help from a teacher or tutor if you are struggling</li> <li>Understanding the real-world applications of simplifying expressions with negative exponents</li> </ul> <h2><strong>Q: What are some additional resources for learning about simplifying expressions with negative exponents?</strong></h2> <hr> <p>A: Some additional resources for learning about simplifying expressions with negative exponents include:</p> <ul> <li>Online tutorials and videos</li> <li>Practice tests and exercises</li> <li>Math textbooks and workbooks</li> <li>Online communities and forums for math students and professionals</li> </ul>