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Chapter 2: Problem 31

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

Short Answer

Expert verified
The simplified expression is \(x^{14} y^{28}\).

Step by step solution

01

Apply the power property (a^m)^n = a^(mn)

We need to distribute the exponent of 2 outside the brackets to both terms inside the brackets: \[ \left(\frac{\left(x^{2} y^{-5}\right)^{-4}}{\left(x^{5} y^{-2}\right)^{-3}}\right)^{2} = \frac{\left(x^{2} y^{-5}\right)^{-8}}{\left(x^{5} y^{-2}\right)^{-6}} \]
02

Distribute the exponents to respective terms

Now we will distribute the exponents, -8 and -6, to their respective terms inside the brackets: \[ \frac{\left(x^{2} y^{-5}\right)^{-8}}{\left(x^{5} y^{-2}\right)^{-6}} = \frac{x^{-16} y^{40}}{x^{-30} y^{12}} \]
03

Use the property a^m / a^n = a^(m-n) to simplify the expression

Now, we will use the power property a^m / a^n = a^(m-n) to simplify the expression: \[ \frac{x^{-16} y^{40}}{x^{-30} y^{12}} = x^{-16 - (-30)}y^{40 - 12} = x^{14} y^{28} \] The simplified expression is: \[ x^{14} y^{28} \]

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Most popular questions from this chapter

Write the indicated expression as a ratio of polynomials, assuming that $$ r(x)=\frac{3 x+4}{x^{2}+1}, \quad s(x)=\frac{x^{2}+2}{2 x-1}, \quad t(x)=\frac{5}{4 x^{3}+3} $$. $$ (r(x))^{2} $$

Suppose \(M\) and \(N\) are odd integers. Explain why $$ x^{2}+M x+N $$ has no integer zeros.

Write each expression as the sum of a polynomial and a rational function whose numerator has smaller degree than its denominator. $$ \frac{x^{2}}{4 x+3} $$

Explain why the composition of a polynomial and a rational function (in either order) is a rational function.

Write each expression as the sum of a polynomial and a rational function whose numerator has smaller degree than its denominator. $$ \frac{x^{6}+3 x^{3}+1}{x^{2}+2 x+5} $$
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