Simplify x^9y^4(4x^-1y^2)^3

$x^{9} y^{4} {(4 x^{- 1} y^{2})}^{3}$

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

$x^{9} y^{4} {(4 \frac{1}{x} y^{2})}^{3}$

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

$x^{9} y^{4} {(\frac{4}{x} y^{2})}^{3}$

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

$x^{9} y^{4} {(\frac{4 y^{2}}{x})}^{3}$

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

Apply the product rule to $\frac{4 y^{2}}{x}$ .

$x^{9} y^{4} \frac{{(4 y^{2})}^{3}}{x^{3}}$

Apply the product rule to $4 y^{2}$ .

$x^{9} y^{4} \frac{4^{3} {(y^{2})}^{3}}{x^{3}}$

Simplify the numerator.

Raise $4$ to the power of $3$ .

$x^{9} y^{4} \frac{64 {(y^{2})}^{3}}{x^{3}}$

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

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

$x^{9} y^{4} \frac{64 y^{2 \cdot 3}}{x^{3}}$

Multiply $2$ by $3$ .

$x^{9} y^{4} \frac{64 y^{6}}{x^{3}}$

Cancel the common factor of $x^{3}$ .

Factor $x^{3}$ out of $x^{9} y^{4}$ .

$x^{3} (x^{6} y^{4}) \frac{64 y^{6}}{x^{3}}$

Cancel the common factor.

$x^{3} (x^{6} y^{4}) \frac{64 y^{6}}{x^{3}}$

Rewrite the expression.

$x^{6} y^{4} (64 y^{6})$

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

$x^{6} (64 y^{4 + 6})$

Add $4$ and $6$ .

$x^{6} (64 y^{10})$

Rewrite using the commutative property of multiplication.

$64 x^{6} y^{10}$