Find The Value Of The Following Expression If \[$ X=2 \$\], \[$ Y=3 \$\], \[$ M=4 \$\], And \[$ N=1 \$\]:$\[ \frac{x^{m+n} \cdot Y^{m-n}}{x^{m-n} \cdot Y^{m+n}} \\]Ans: \[$\frac{4}{9}\$\]

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Introduction

In mathematics, simplifying expressions is a crucial skill that helps us solve complex problems and arrive at accurate solutions. Exponential expressions, in particular, can be challenging to simplify, but with the right approach, we can break them down and find their values. In this article, we will explore how to simplify the expression xm+n⋅ym−nxm−n⋅ym+n\frac{x^{m+n} \cdot y^{m-n}}{x^{m-n} \cdot y^{m+n}} given the values of xx, yy, mm, and nn. We will use the given values x=2x=2, y=3y=3, m=4m=4, and n=1n=1 to find the value of the expression.

Understanding Exponential Expressions

Exponential expressions involve variables raised to powers. In the given expression, we have xx raised to the power of m+nm+n, yy raised to the power of m−nm-n, xx raised to the power of m−nm-n, and yy raised to the power of m+nm+n. To simplify this expression, we need to apply the rules of exponents.

Applying the Rules of Exponents

The rules of exponents state that when we multiply two exponential expressions with the same base, we add their exponents. Similarly, when we divide two exponential expressions with the same base, we subtract their exponents. We can use these rules to simplify the given expression.

Simplifying the Expression

Let's start by simplifying the numerator of the expression:

xm+n⋅ym−nxm−n⋅ym+n=xm+n⋅ym−nxm−n⋅ym+n\frac{x^{m+n} \cdot y^{m-n}}{x^{m-n} \cdot y^{m+n}} = \frac{x^{m+n} \cdot y^{m-n}}{x^{m-n} \cdot y^{m+n}}

We can rewrite the numerator as:

xm+n⋅ym−n=xm+n⋅ym−nx^{m+n} \cdot y^{m-n} = x^{m+n} \cdot y^{m-n}

Now, we can apply the rule of exponents to simplify the numerator:

xm+n⋅ym−n=xm+n⋅ym−n=xm+n⋅ym−nx^{m+n} \cdot y^{m-n} = x^{m+n} \cdot y^{m-n} = x^{m+n} \cdot y^{m-n}

Similarly, we can rewrite the denominator as:

xm−n⋅ym+n=xm−n⋅ym+nx^{m-n} \cdot y^{m+n} = x^{m-n} \cdot y^{m+n}

Now, we can apply the rule of exponents to simplify the denominator:

xm−n⋅ym+n=xm−n⋅ym+n=xm−n⋅ym+nx^{m-n} \cdot y^{m+n} = x^{m-n} \cdot y^{m+n} = x^{m-n} \cdot y^{m+n}

Dividing the Numerator and Denominator

Now that we have simplified the numerator and denominator, we can divide them to find the value of the expression:

xm+n⋅ym−nxm−n⋅ym+n=xm+n⋅ym−nxm−n⋅ym+n\frac{x^{m+n} \cdot y^{m-n}}{x^{m-n} \cdot y^{m+n}} = \frac{x^{m+n} \cdot y^{m-n}}{x^{m-n} \cdot y^{m+n}}

We can rewrite the expression as:

xm+n⋅ym−nxm−n⋅ym+n=xm+nxm−n⋅ym−nym+n\frac{x^{m+n} \cdot y^{m-n}}{x^{m-n} \cdot y^{m+n}} = \frac{x^{m+n}}{x^{m-n}} \cdot \frac{y^{m-n}}{y^{m+n}}

Now, we can apply the rule of exponents to simplify the expression:

xm+nxm−n=xm+n−m+n=x2n\frac{x^{m+n}}{x^{m-n}} = x^{m+n-m+n} = x^{2n}

ym−nym+n=ym−n−m−n=y−2n\frac{y^{m-n}}{y^{m+n}} = y^{m-n-m-n} = y^{-2n}

Substituting the Values of x, y, m, and n

Now that we have simplified the expression, we can substitute the values of xx, yy, mm, and nn to find the value of the expression:

x=2x = 2

y=3y = 3

m=4m = 4

n=1n = 1

Substituting these values into the expression, we get:

xm+n⋅ym−nxm−n⋅ym+n=24+1⋅34−124−1⋅34+1\frac{x^{m+n} \cdot y^{m-n}}{x^{m-n} \cdot y^{m+n}} = \frac{2^{4+1} \cdot 3^{4-1}}{2^{4-1} \cdot 3^{4+1}}

Simplifying the expression, we get:

25⋅3323⋅35=32⋅278⋅243\frac{2^{5} \cdot 3^{3}}{2^{3} \cdot 3^{5}} = \frac{32 \cdot 27}{8 \cdot 243}

Final Answer

Simplifying the expression further, we get:

32⋅278⋅243=8641944=49\frac{32 \cdot 27}{8 \cdot 243} = \frac{864}{1944} = \frac{4}{9}

Therefore, the value of the expression is 49\frac{4}{9}.

Conclusion

In this article, we have learned how to simplify exponential expressions using the rules of exponents. We have applied these rules to simplify the expression xm+n⋅ym−nxm−n⋅ym+n\frac{x^{m+n} \cdot y^{m-n}}{x^{m-n} \cdot y^{m+n}} given the values of xx, yy, mm, and nn. We have used the given values x=2x=2, y=3y=3, m=4m=4, and n=1n=1 to find the value of the expression, which is 49\frac{4}{9}. This problem demonstrates the importance of understanding the rules of exponents and how to apply them to simplify complex expressions.

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Introduction

In our previous article, we explored how to simplify exponential expressions using the rules of exponents. We applied these rules to simplify the expression xm+n⋅ym−nxm−n⋅ym+n\frac{x^{m+n} \cdot y^{m-n}}{x^{m-n} \cdot y^{m+n}} given the values of xx, yy, mm, and nn. In this article, we will answer some frequently asked questions about simplifying exponential expressions.

Q: What are the rules of exponents?

A: The rules of exponents state that when we multiply two exponential expressions with the same base, we add their exponents. Similarly, when we divide two exponential expressions with the same base, we subtract their exponents.

Q: How do I simplify an exponential expression?

A: To simplify an exponential expression, you need to apply the rules of exponents. If the expression involves multiplication, you add the exponents. If the expression involves division, you subtract the exponents.

Q: What is the difference between xmx^m and xnx^n?

A: xmx^m and xnx^n are two different exponential expressions with the same base xx. The exponent mm is the power to which xx is raised in the first expression, while the exponent nn is the power to which xx is raised in the second expression.

Q: How do I simplify an expression like xmxn\frac{x^m}{x^n}?

A: To simplify an expression like xmxn\frac{x^m}{x^n}, you need to subtract the exponents. This is because the expression involves division, and the rule of exponents states that when we divide two exponential expressions with the same base, we subtract their exponents.

Q: What is the value of xm⋅ynxn⋅ym\frac{x^m \cdot y^n}{x^n \cdot y^m}?

A: To simplify this expression, you need to apply the rules of exponents. You can rewrite the expression as xmxn⋅ynym\frac{x^m}{x^n} \cdot \frac{y^n}{y^m}. Then, you can simplify each fraction separately. The first fraction simplifies to xm−nx^{m-n}, and the second fraction simplifies to yn−my^{n-m}. Therefore, the value of the expression is xm−nyn−m\frac{x^{m-n}}{y^{n-m}}.

Q: How do I simplify an expression like xm+nxm−n\frac{x^{m+n}}{x^{m-n}}?

A: To simplify this expression, you need to apply the rules of exponents. You can rewrite the expression as xm⋅xnxm⋅x−n\frac{x^m \cdot x^n}{x^m \cdot x^{-n}}. Then, you can simplify each fraction separately. The first fraction simplifies to xnx^n, and the second fraction simplifies to x−nx^{-n}. Therefore, the value of the expression is xn+n=x2nx^{n+n} = x^{2n}.

Q: What is the value of 24⋅3323⋅35\frac{2^4 \cdot 3^3}{2^3 \cdot 3^5}?

A: To simplify this expression, you need to apply the rules of exponents. You can rewrite the expression as 2423⋅3335\frac{2^4}{2^3} \cdot \frac{3^3}{3^5}. Then, you can simplify each fraction separately. The first fraction simplifies to 24−3=212^{4-3} = 2^1, and the second fraction simplifies to 33−5=3−23^{3-5} = 3^{-2}. Therefore, the value of the expression is 21⋅3−2=292^1 \cdot 3^{-2} = \frac{2}{9}.

Conclusion

In this article, we have answered some frequently asked questions about simplifying exponential expressions. We have applied the rules of exponents to simplify various expressions and have provided examples to illustrate the concepts. By understanding the rules of exponents and how to apply them, you can simplify complex expressions and arrive at accurate solutions.