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Chapter 2: Problem 32

Simplify the given expression. $$ \left(\frac{\left(x^{-3} y^{5}\right)^{-4}}{\left(x^{-5} y^{-2}\right)^{-3}}\right)^{-2} $$

Short Answer

Expert verified
The simplified form of the given expression is \( \frac{ y^{40} }{ x^6} \).

Step by step solution

01

Rewriting Expression

First, we can rewrite the given expression as: \[ \left( \frac{ (x^{-3} y^{5})^{-4} }{ (x^{-5} y^{-2})^{-3} } \right)^{-2} \]
02

Apply Power of a Quotient Rule

Apply the Power of a Quotient property to redistribute power to numerator and denominator, we have: \[ \frac{ (x^{-3} y^{5})^{-4(-2)} }{ (x^{-5} y^{-2})^{-3(-2)} } \]
03

Simplify the Exponents

Now, simplify the exponents in the expression: \[ \frac{ (x^{-3} y^{5})^{8} }{ (x^{-5} y^{-2})^{6} } \]
04

Use Power of a Product Rule

Apply the Power of a Product rule to expand exponents: \[ \frac{x^{-3(8)} y^{5(8)}}{x^{-5(6)} y^{-2(6)}} \]
05

Simplify the Exponents Again

Simply the exponents of x and y again: \[ \frac{x^{-24} y^{40}}{x^{-30} y^{-12}} \]
06

Combining Like Terms

Combine like terms, adding exponents of matching bases within each respective set (numerator and denominator): \[ \frac{ y^{40} }{ x^6} \] The simplified form of the given expression is \( \frac{ y^{40} }{ x^6} \).

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Most popular questions from this chapter

Find the asymptotes of the graph of the given function \(\mathrm{r}\). $$ r(x)=\frac{9 x+5}{x^{2}-x-6} $$

Write the indicated expression as \(a\) polynomial. $$ \frac{s(1+x)-s(1)}{x} $$

Suppose \(p\) and \(q\) are polynomials of degree 3 such that \(p(1)=q(1), p(2)=q(2), p(3)=q(3),\) and \(p(4)=q(4) .\) Explain why \(p=q\).

Verify that \((x+y)^{3}=x^{3}+3 x^{2} y+3 x y^{2}+y^{3}\).

Find the asymptotes of the graph of the given function \(\mathrm{r}\). $$ r(x)=\frac{6 x^{4}+4 x^{3}-7}{2 x^{4}+3 x^{2}+5} $$
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