Simplify (5x^7y^-1)^-2

${(5 x^{7} y^{- 1})}^{- 2}$

Step 1

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

${(5 x^{7} \frac{1}{y})}^{- 2}$

Step 2

Multiply $5 x^{7} \frac{1}{y}$ .

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

${(x^{7} \frac{5}{y})}^{- 2}$

Combine $x^{7}$ and $\frac{5}{y}$ .

${(\frac{x^{7} \cdot 5}{y})}^{- 2}$

Step 3

Move $5$ to the left of $x^{7}$ .

${(\frac{5 \cdot x^{7}}{y})}^{- 2}$

Step 4

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

${(\frac{y}{5 x^{7}})}^{2}$

Step 5

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

Apply the product rule to $\frac{y}{5 x^{7}}$ .

$\frac{y^{2}}{{(5 x^{7})}^{2}}$

Apply the product rule to $5 x^{7}$ .

$\frac{y^{2}}{5^{2} {(x^{7})}^{2}}$

Step 6

Simplify the denominator.

Raise $5$ to the power of $2$ .

$\frac{y^{2}}{25 {(x^{7})}^{2}}$

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

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

$\frac{y^{2}}{25 x^{7 \cdot 2}}$

Multiply $7$ by $2$ .

$\frac{y^{2}}{25 x^{14}}$