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

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

Step 1

Move $x^{- 2}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 2

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

Move $x^{2}$ .

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

Multiply $x^{2}$ by $x$ .

Raise $x$ to the power of $1$ .

${(\frac{4 y^{3}}{x^{2} x^{1} y^{- 4}})}^{- 2}$

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

${(\frac{4 y^{3}}{x^{2 + 1} y^{- 4}})}^{- 2}$

Add $2$ and $1$ .

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

Step 3

Move $y^{- 4}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

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

Step 4

Multiply $y^{3}$ by $y^{4}$ by adding the exponents.

Move $y^{4}$ .

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

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

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

Add $4$ and $3$ .

${(\frac{4 y^{7}}{x^{3}})}^{- 2}$

Step 5

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

${(\frac{x^{3}}{4 y^{7}})}^{2}$

Step 6

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

Apply the product rule to $\frac{x^{3}}{4 y^{7}}$ .

$\frac{{(x^{3})}^{2}}{{(4 y^{7})}^{2}}$

Apply the product rule to $4 y^{7}$ .

$\frac{{(x^{3})}^{2}}{4^{2} {(y^{7})}^{2}}$

Step 7

Multiply the exponents in ${(x^{3})}^{2}$ .

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

$\frac{x^{3 \cdot 2}}{4^{2} {(y^{7})}^{2}}$

Multiply $3$ by $2$ .

$\frac{x^{6}}{4^{2} {(y^{7})}^{2}}$

Step 8

Simplify the denominator.

Raise $4$ to the power of $2$ .

$\frac{x^{6}}{16 {(y^{7})}^{2}}$

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

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

$\frac{x^{6}}{16 y^{7 \cdot 2}}$

Multiply $7$ by $2$ .

$\frac{x^{6}}{16 y^{14}}$

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

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