Simplify (x^-2y^-4x^3)^-2

${(x^{- 2} y^{- 4} x^{3})}^{- 2}$

Step 1

Multiply $x^{- 2}$ by $x^{3}$ by adding the exponents.

Move $x^{3}$ .

${(x^{3} x^{- 2} y^{- 4})}^{- 2}$

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

${(x^{3 - 2} y^{- 4})}^{- 2}$

Subtract $2$ from $3$ .

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

Step 2

Simplify $x^{1} y^{- 4}$ .

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

Step 3

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

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

Step 4

Combine $x$ and $\frac{1}{y^{4}}$ .

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

Step 5

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

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

Step 6

Apply the product rule to $\frac{y^{4}}{x}$ .

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

Step 7

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

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

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

Multiply $4$ by $2$ .

$\frac{y^{8}}{x^{2}}$