Simplify (-5x^2y^-1)^4

${(- 5 x^{2} y^{- 1})}^{4}$

Step 1

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

${(- 5 x^{2} \frac{1}{y})}^{4}$

Step 2

Multiply $- 5 x^{2} \frac{1}{y}$ .

Combine $- 5$ and $\frac{1}{y}$ .

${(x^{2} \frac{- 5}{y})}^{4}$

Combine $x^{2}$ and $\frac{- 5}{y}$ .

${(\frac{x^{2} \cdot - 5}{y})}^{4}$

Step 3

Move $- 5$ to the left of $x^{2}$ .

${(\frac{- 5 \cdot x^{2}}{y})}^{4}$

Step 4

Move the negative in front of the fraction.

${(- \frac{(5) \cdot x^{2}}{y})}^{4}$

Step 5

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

Apply the product rule to $- \frac{5 x^{2}}{y}$ .

${(- 1)}^{4} {(\frac{5 x^{2}}{y})}^{4}$

Apply the product rule to $\frac{5 x^{2}}{y}$ .

${(- 1)}^{4} \frac{{(5 x^{2})}^{4}}{y^{4}}$

Apply the product rule to $5 x^{2}$ .

${(- 1)}^{4} \frac{5^{4} {(x^{2})}^{4}}{y^{4}}$

Step 6

Simplify the expression.

Raise $- 1$ to the power of $4$ .

$1 \frac{5^{4} {(x^{2})}^{4}}{y^{4}}$

Multiply $\frac{5^{4} {(x^{2})}^{4}}{y^{4}}$ by $1$ .

$\frac{5^{4} {(x^{2})}^{4}}{y^{4}}$

Step 7

Simplify the numerator.

Raise $5$ to the power of $4$ .

$\frac{625 {(x^{2})}^{4}}{y^{4}}$

Multiply the exponents in ${(x^{2})}^{4}$ .

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

$\frac{625 x^{2 \cdot 4}}{y^{4}}$

Multiply $2$ by $4$ .

$\frac{625 x^{8}}{y^{4}}$