Simplify (4x^-2y^4)/(8x^3(y^-2)^3)

$\frac{4 x^{- 2} y^{4}}{8 x^{3} {(y^{- 2})}^{3}}$

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

$\frac{4 y^{4}}{8 x^{3} {(y^{- 2})}^{3} x^{2}}$

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

Move $x^{2}$ .

$\frac{4 y^{4}}{8 (x^{2} x^{3}) {(y^{- 2})}^{3}}$

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

$\frac{4 y^{4}}{8 x^{2 + 3} {(y^{- 2})}^{3}}$

Add $2$ and $3$ .

$\frac{4 y^{4}}{8 x^{5} {(y^{- 2})}^{3}}$

Cancel the common factor of $4$ and $8$ .

Factor $4$ out of $4 y^{4}$ .

$\frac{4 (y^{4})}{8 x^{5} {(y^{- 2})}^{3}}$

Cancel the common factors.

Factor $4$ out of $8 x^{5} {(y^{- 2})}^{3}$ .

$\frac{4 (y^{4})}{4 (2 x^{5} {(y^{- 2})}^{3})}$

Cancel the common factor.

$\frac{4 y^{4}}{4 (2 x^{5} {(y^{- 2})}^{3})}$

Rewrite the expression.

$\frac{y^{4}}{2 x^{5} {(y^{- 2})}^{3}}$

Simplify the denominator.

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

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

$\frac{y^{4}}{2 x^{5} y^{- 2 \cdot 3}}$

Multiply $- 2$ by $3$ .

$\frac{y^{4}}{2 x^{5} y^{- 6}}$

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

$\frac{y^{4}}{2 x^{5} \frac{1}{y^{6}}}$

Combine exponents.

Combine $2$ and $\frac{1}{y^{6}}$ .

$\frac{y^{4}}{x^{5} \frac{2}{y^{6}}}$

Combine $x^{5}$ and $\frac{2}{y^{6}}$ .

$\frac{y^{4}}{\frac{x^{5} \cdot 2}{y^{6}}}$

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

$\frac{y^{4}}{\frac{2 x^{5}}{y^{6}}}$

Multiply the numerator by the reciprocal of the denominator.

$y^{4} \frac{y^{6}}{2 x^{5}}$

Multiply $y^{4} \frac{y^{6}}{2 x^{5}}$ .

Combine $y^{4}$ and $\frac{y^{6}}{2 x^{5}}$ .

$\frac{y^{4} y^{6}}{2 x^{5}}$

Multiply $y^{4}$ by $y^{6}$ by adding the exponents.

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

$\frac{y^{4 + 6}}{2 x^{5}}$

Add $4$ and $6$ .

$\frac{y^{10}}{2 x^{5}}$