Simplify 8x^6y(2x^-1y)^3

$8 x^{6} y {(2 x^{- 1} y)}^{3}$

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

$8 x^{6} y {(2 \frac{1}{x} y)}^{3}$

Combine $2$ and $\frac{1}{x}$ .

$8 x^{6} y {(\frac{2}{x} y)}^{3}$

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

$8 x^{6} y {(\frac{2 y}{x})}^{3}$

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

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

$8 x^{6} y \frac{{(2 y)}^{3}}{x^{3}}$

Apply the product rule to $2 y$ .

$8 x^{6} y \frac{2^{3} y^{3}}{x^{3}}$

Raise $2$ to the power of $3$ .

$8 x^{6} y \frac{8 y^{3}}{x^{3}}$

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

Factor $x^{3}$ out of $8 x^{6} y$ .

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

Cancel the common factor.

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

Rewrite the expression.

$8 x^{3} y (8 y^{3})$

Multiply $8$ by $8$ .

$64 x^{3} y \cdot y^{3}$

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

Move $y^{3}$ .

$64 x^{3} (y^{3} y)$

Multiply $y^{3}$ by $y$ .

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

$64 x^{3} (y^{3} y^{1})$

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

$64 x^{3} y^{3 + 1}$

Add $3$ and $1$ .

$64 x^{3} y^{4}$