Simplify ((-5a^5b^3)/(10a^7b^2))^-2

${(\frac{- 5 a^{5} b^{3}}{10 a^{7} b^{2}})}^{- 2}$

Cancel the common factor of $- 5$ and $10$ .

Factor $5$ out of $- 5 a^{5} b^{3}$ .

${(\frac{5 (- a^{5} b^{3})}{10 a^{7} b^{2}})}^{- 2}$

Cancel the common factors.

Factor $5$ out of $10 a^{7} b^{2}$ .

${(\frac{5 (- a^{5} b^{3})}{5 (2 a^{7} b^{2})})}^{- 2}$

Cancel the common factor.

${(\frac{5 (- a^{5} b^{3})}{5 (2 a^{7} b^{2})})}^{- 2}$

Rewrite the expression.

${(\frac{- a^{5} b^{3}}{2 a^{7} b^{2}})}^{- 2}$

Cancel the common factor of $a^{5}$ and $a^{7}$ .

Factor $a^{5}$ out of $- a^{5} b^{3}$ .

${(\frac{a^{5} (- 1 b^{3})}{2 a^{7} b^{2}})}^{- 2}$

Cancel the common factors.

Factor $a^{5}$ out of $2 a^{7} b^{2}$ .

${(\frac{a^{5} (- 1 b^{3})}{a^{5} (2 a^{2} b^{2})})}^{- 2}$

Cancel the common factor.

${(\frac{a^{5} (- 1 b^{3})}{a^{5} (2 a^{2} b^{2})})}^{- 2}$

Rewrite the expression.

${(\frac{- 1 b^{3}}{2 a^{2} b^{2}})}^{- 2}$

Cancel the common factor of $b^{3}$ and $b^{2}$ .

Factor $b^{2}$ out of $- 1 b^{3}$ .

${(\frac{b^{2} (- 1 b)}{2 a^{2} b^{2}})}^{- 2}$

Cancel the common factors.

Factor $b^{2}$ out of $2 a^{2} b^{2}$ .

${(\frac{b^{2} (- 1 b)}{b^{2} (2 a^{2})})}^{- 2}$

Cancel the common factor.

${(\frac{b^{2} (- 1 b)}{b^{2} (2 a^{2})})}^{- 2}$

Rewrite the expression.

${(\frac{- 1 b}{2 a^{2}})}^{- 2}$

Move the negative in front of the fraction.

${(- \frac{b}{2 a^{2}})}^{- 2}$

Change the sign of the exponent by rewriting the base as its reciprocal.

${(- \frac{2 a^{2}}{b})}^{2}$

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

Apply the product rule to $- \frac{2 a^{2}}{b}$ .

${(- 1)}^{2} {(\frac{2 a^{2}}{b})}^{2}$

Apply the product rule to $\frac{2 a^{2}}{b}$ .

${(- 1)}^{2} \frac{{(2 a^{2})}^{2}}{b^{2}}$

Apply the product rule to $2 a^{2}$ .

${(- 1)}^{2} \frac{2^{2} {(a^{2})}^{2}}{b^{2}}$

Simplify the expression.

Raise $- 1$ to the power of $2$ .

$1 \frac{2^{2} {(a^{2})}^{2}}{b^{2}}$

Multiply $\frac{2^{2} {(a^{2})}^{2}}{b^{2}}$ by $1$ .

$\frac{2^{2} {(a^{2})}^{2}}{b^{2}}$

Simplify the numerator.

Raise $2$ to the power of $2$ .

$\frac{4 {(a^{2})}^{2}}{b^{2}}$

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

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

$\frac{4 a^{2 \cdot 2}}{b^{2}}$

Multiply $2$ by $2$ .

$\frac{4 a^{4}}{b^{2}}$

Simplify ((-5a^5b^3)/(10a^7b^2))^-2

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