Simplify ((20x^-3)/(10x^-1))^-2

${(\frac{20 x^{- 3}}{10 x^{- 1}})}^{- 2}$

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

${(\frac{20}{10 x^{- 1} x^{3}})}^{- 2}$

Multiply $x^{- 1}$ by $x^{3}$ by adding the exponents.

Move $x^{3}$ .

${(\frac{20}{10 (x^{3} x^{- 1})})}^{- 2}$

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

${(\frac{20}{10 x^{3 - 1}})}^{- 2}$

Subtract $1$ from $3$ .

${(\frac{20}{10 x^{2}})}^{- 2}$

Cancel the common factor of $20$ and $10$ .

Factor $10$ out of $20$ .

${(\frac{10 \cdot 2}{10 x^{2}})}^{- 2}$

Cancel the common factors.

Factor $10$ out of $10 x^{2}$ .

${(\frac{10 \cdot 2}{10 (x^{2})})}^{- 2}$

Cancel the common factor.

${(\frac{10 \cdot 2}{10 x^{2}})}^{- 2}$

Rewrite the expression.

${(\frac{2}{x^{2}})}^{- 2}$

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

${(\frac{x^{2}}{2})}^{2}$

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

$\frac{{(x^{2})}^{2}}{2^{2}}$

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

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

$\frac{x^{2 \cdot 2}}{2^{2}}$

Multiply $2$ by $2$ .

$\frac{x^{4}}{2^{2}}$

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

$\frac{x^{4}}{4}$