Simplify (((x^2y^3)^-1)/((x^-2y^2z)^2))^2

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

Step 1

Move ${(x^{2} y^{3})}^{- 1}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 2

Simplify the denominator.

Apply the product rule to $x^{- 2} y^{2} z$ .

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

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

${(\frac{1}{{(\frac{1}{x^{2}} y^{2})}^{2} z^{2} x^{2} y^{3}})}^{2}$

Combine $\frac{1}{x^{2}}$ and $y^{2}$ .

${(\frac{1}{{(\frac{y^{2}}{x^{2}})}^{2} z^{2} x^{2} y^{3}})}^{2}$

Apply the product rule to $\frac{y^{2}}{x^{2}}$ .

${(\frac{1}{\frac{{(y^{2})}^{2}}{{(x^{2})}^{2}} z^{2} x^{2} y^{3}})}^{2}$

Multiply the exponents in ${(y^{2})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(\frac{1}{\frac{y^{2 \cdot 2}}{{(x^{2})}^{2}} z^{2} x^{2} y^{3}})}^{2}$

Multiply $2$ by $2$ .

${(\frac{1}{\frac{y^{4}}{{(x^{2})}^{2}} z^{2} x^{2} y^{3}})}^{2}$

Multiply the exponents in ${(x^{2})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(\frac{1}{\frac{y^{4}}{x^{2 \cdot 2}} z^{2} x^{2} y^{3}})}^{2}$

Multiply $2$ by $2$ .

${(\frac{1}{\frac{y^{4}}{x^{4}} z^{2} x^{2} y^{3}})}^{2}$

Combine exponents.

Combine $\frac{y^{4}}{x^{4}}$ and $z^{2}$ .

${(\frac{1}{\frac{y^{4} z^{2}}{x^{4}} x^{2} y^{3}})}^{2}$

Combine $\frac{y^{4} z^{2}}{x^{4}}$ and $x^{2}$ .

${(\frac{1}{\frac{y^{4} z^{2} x^{2}}{x^{4}} y^{3}})}^{2}$

Combine $\frac{y^{4} z^{2} x^{2}}{x^{4}}$ and $y^{3}$ .

${(\frac{1}{\frac{y^{4} z^{2} x^{2} y^{3}}{x^{4}}})}^{2}$

Multiply $y^{4}$ by $y^{3}$ by adding the exponents.

Move $y^{3}$ .

${(\frac{1}{\frac{y^{3} y^{4} z^{2} x^{2}}{x^{4}}})}^{2}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

${(\frac{1}{\frac{y^{3 + 4} z^{2} x^{2}}{x^{4}}})}^{2}$

Add $3$ and $4$ .

${(\frac{1}{\frac{y^{7} z^{2} x^{2}}{x^{4}}})}^{2}$

Reduce the expression $\frac{y^{7} z^{2} x^{2}}{x^{4}}$ by cancelling the common factors.

Factor $x^{2}$ out of $y^{7} z^{2} x^{2}$ .

${(\frac{1}{\frac{x^{2} (y^{7} z^{2})}{x^{4}}})}^{2}$

Factor $x^{2}$ out of $x^{4}$ .

${(\frac{1}{\frac{x^{2} (y^{7} z^{2})}{x^{2} x^{2}}})}^{2}$

Cancel the common factor.

${(\frac{1}{\frac{x^{2} (y^{7} z^{2})}{x^{2} x^{2}}})}^{2}$

Rewrite the expression.

${(\frac{1}{\frac{y^{7} z^{2}}{x^{2}}})}^{2}$

Step 3

Multiply the numerator by the reciprocal of the denominator.

${(1 \frac{x^{2}}{y^{7} z^{2}})}^{2}$

Step 4

Multiply $\frac{x^{2}}{y^{7} z^{2}}$ by $1$ .

${(\frac{x^{2}}{y^{7} z^{2}})}^{2}$

Step 5

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $\frac{x^{2}}{y^{7} z^{2}}$ .

$\frac{{(x^{2})}^{2}}{{(y^{7} z^{2})}^{2}}$

Apply the product rule to $y^{7} z^{2}$ .

$\frac{{(x^{2})}^{2}}{{(y^{7})}^{2} {(z^{2})}^{2}}$

Step 6

Multiply the exponents in ${(x^{2})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{x^{2 \cdot 2}}{{(y^{7})}^{2} {(z^{2})}^{2}}$

Multiply $2$ by $2$ .

$\frac{x^{4}}{{(y^{7})}^{2} {(z^{2})}^{2}}$

Step 7

Simplify the denominator.

Multiply the exponents in ${(y^{7})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{x^{4}}{y^{7 \cdot 2} {(z^{2})}^{2}}$

Multiply $7$ by $2$ .

$\frac{x^{4}}{y^{14} {(z^{2})}^{2}}$

Multiply the exponents in ${(z^{2})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{x^{4}}{y^{14} z^{2 \cdot 2}}$

Multiply $2$ by $2$ .

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

Simplify (((x^2y^3)^-1)/((x^-2y^2z)^2))^2

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