Question

Simplify the given expression using only positive exponents. Then complete the statements that follow. [(X^2y^3)^-1/(x^-2y^2z)^2]^2
The exponent on x is___
The exponent on y is___
The exponent on a is____

Answer 1

Answer:

As

$\:\left(\frac{\left(x^2y^3\right)^{-1}}{\left(x^{-2}y^2z\right)^2}\right)^2=x^4y^{-14}z^{-4}$

• The exponent on x is 4

• The exponent on y is -14

• The exponent on z is -4

Step-by-step explanation:

Given the expression

$\left[\frac{\left(x^2y^3\right)^{-1}}{\left(x^{-2}y^2z\right)^2}\right]^2$

$\mathrm{Apply\:exponent\:rule}:\quad \left(\frac{a}{b}\right)^c=\frac{a^c}{b^c}$

$\left(\frac{\left(x^2y^3\right)^{-1}}{\left(x^{-2}y^2z\right)^2}\right)^2=\frac{\left(\left(x^2y^3\right)^{-1}\right)^2}{\left(\left(x^{-2}y^2z\right)^2\right)^2}$

                      $=\frac{\left(\left(x^2y^3\right)^{-1}\right)^2}{\left(\left(x^{-2}y^2z\right)^2\right)^2}$

as

$\mathrm{Apply\:exponent\:rule}:\quad \:a^{-1}=\frac{1}{a}$

so the expression becomes

                       $=\frac{\frac{1}{x^4y^6}}{\left(\left(x^{-2}y^2z\right)^2\right)^2}$       ∵ $\left(x^2y^3\right)^{-1}=\frac{1}{x^2y^3}$

as

$\mathrm{Apply\:exponent\:rule}:\quad \left(a\cdot \:b\right)^n=a^nb^n$

so the expression becomes

                      $=\frac{\frac{1}{x^4y^6}}{\frac{y^8z^4}{x^8}}$                    ∵ $\left(x^{-2}y^2z\right)^2=\frac{y^4z^2}{x^4}$

as

$\mathrm{Divide\:fractions}:\quad \frac{\frac{a}{b}}{\frac{c}{d}}=\frac{a\cdot \:d}{b\cdot \:c}$

so the expression becomes

                          $=\frac{1\cdot \:x^8}{x^4y^6y^8z^4}$

                         $=\frac{x^8}{x^4y^6y^8z^4}$

as

$\mathrm{Apply\:exponent\:rule}:\quad \frac{x^a}{x^b}=x^{a-b}$

so the expression becomes

                        $=\frac{x^{8-4}}{y^8y^6z^4}$

                        $=\frac{x^4}{y^8y^6z^4}$

                        $=\frac{x^4}{y^{14}z^4}$

as

$\mathrm{Apply\:exponent\:rule}:\quad \:a^{-1}=\frac{1}{a}$

so the expression becomes

                           $=x^4y^{-14}z^{-4}$

Therefore,

• The exponent on x is 4

• The exponent on y is -14

• The exponent on z is -4

Answer 2

Answer:

x=4

y=14

z=4

Step-by-step explanation: