Evaluate ((-6x^2)^3)/((3x^-4)^2)

$\frac{{(- 6 x^{2})}^{3}}{{(3 x^{- 4})}^{2}}$

Simplify the numerator.

Apply the product rule to $- 6 x^{2}$ .

$\frac{{(- 6)}^{3} {(x^{2})}^{3}}{{(3 x^{- 4})}^{2}}$

Raise $- 6$ to the power of $3$ .

$\frac{- 216 {(x^{2})}^{3}}{{(3 x^{- 4})}^{2}}$

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

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

$\frac{- 216 x^{2 \cdot 3}}{{(3 x^{- 4})}^{2}}$

Multiply $2$ by $3$ .

$\frac{- 216 x^{6}}{{(3 x^{- 4})}^{2}}$

Simplify the denominator.

Apply the product rule to $3 x^{- 4}$ .

$\frac{- 216 x^{6}}{3^{2} {(x^{- 4})}^{2}}$

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

$\frac{- 216 x^{6}}{9 {(x^{- 4})}^{2}}$

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

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

$\frac{- 216 x^{6}}{9 x^{- 4 \cdot 2}}$

Multiply $- 4$ by $2$ .

$\frac{- 216 x^{6}}{9 x^{- 8}}$

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

$\frac{- 216 x^{6}}{9 \frac{1}{x^{8}}}$

Combine $9$ and $\frac{1}{x^{8}}$ .

$\frac{- 216 x^{6}}{\frac{9}{x^{8}}}$

Multiply the numerator by the reciprocal of the denominator.

$- 216 x^{6} \frac{x^{8}}{9}$

Cancel the common factor of $9$ .

Factor $9$ out of $- 216 x^{6}$ .

$9 (- 24 x^{6}) \frac{x^{8}}{9}$

Cancel the common factor.

$9 (- 24 x^{6}) \frac{x^{8}}{9}$

Rewrite the expression.

$- 24 x^{6} x^{8}$

Multiply $x^{6}$ by $x^{8}$ by adding the exponents.

Move $x^{8}$ .

$- 24 (x^{8} x^{6})$

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

$- 24 x^{8 + 6}$

Add $8$ and $6$ .

$- 24 x^{14}$