Evaluate ( cube root of -6)^3

${(\sqrt[3]{- 6})}^{3}$

Step 1

Rewrite $- 6$ as ${(- 1)}^{3} \cdot 6$ .

Rewrite $- 6$ as $- 1 (6)$ .

${\sqrt[3]{- 1 (6)}}^{3}$

Rewrite $- 1$ as ${(- 1)}^{3}$ .

${\sqrt[3]{{(- 1)}^{3} \cdot 6}}^{3}$

Step 2

Pull terms out from under the radical.

${(- 1 \sqrt[3]{6})}^{3}$

Step 3

Simplify the expression.

Rewrite $- 1 \sqrt[3]{6}$ as $- \sqrt[3]{6}$ .

${(- \sqrt[3]{6})}^{3}$

Apply the product rule to $- \sqrt[3]{6}$ .

${(- 1)}^{3} {\sqrt[3]{6}}^{3}$

Raise $- 1$ to the power of $3$ .

$- {\sqrt[3]{6}}^{3}$

Step 4

Rewrite ${\sqrt[3]{6}}^{3}$ as $6$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{6}$ as $6^{\frac{1}{3}}$ .

$- {(6^{\frac{1}{3}})}^{3}$

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

$- 6^{\frac{1}{3} \cdot 3}$

Combine $\frac{1}{3}$ and $3$ .

$- 6^{\frac{3}{3}}$

Cancel the common factor of $3$ .

Cancel the common factor.

$- 6^{\frac{3}{3}}$

Rewrite the expression.

$- 6^{1}$

Evaluate the exponent.

$- 1 \cdot 6$

Step 5

Multiply $- 1$ by $6$ .

$- 6$