Simplify ( fifth root of -8)^8

${(\sqrt[5]{- 8})}^{8}$

Step 1

Rewrite $- 8$ as ${(- 1)}^{5} \cdot 8$ .

Rewrite $- 8$ as $- 1 (8)$ .

${\sqrt[5]{- 1 (8)}}^{8}$

Rewrite $- 1$ as ${(- 1)}^{5}$ .

${\sqrt[5]{{(- 1)}^{5} \cdot 8}}^{8}$

${\sqrt[5]{{(- 1)}^{5} \cdot 8}}^{8}$

Step 2

Pull terms out from under the radical.

${(- 1 \sqrt[5]{8})}^{8}$

Step 3

Simplify the expression.

Rewrite $- 1 \sqrt[5]{8}$ as $- \sqrt[5]{8}$ .

${(- \sqrt[5]{8})}^{8}$

Apply the product rule to $- \sqrt[5]{8}$ .

${(- 1)}^{8} {\sqrt[5]{8}}^{8}$

Raise $- 1$ to the power of $8$ .

$1 {\sqrt[5]{8}}^{8}$

Multiply ${\sqrt[5]{8}}^{8}$ by $1$ .

${\sqrt[5]{8}}^{8}$

${\sqrt[5]{8}}^{8}$

Step 4

Rewrite ${\sqrt[5]{8}}^{8}$ as ${(8^{8})}^{\frac{1}{5}}$ .

$\sqrt[5]{8^{8}}$

Step 5

Raise $8$ to the power of $8$ .

$\sqrt[5]{16777216}$

Step 6

Rewrite $16777216$ as $16^{5} \cdot 16$ .

Factor $1048576$ out of $16777216$ .

$\sqrt[5]{1048576 (16)}$

Rewrite $1048576$ as $16^{5}$ .

$\sqrt[5]{16^{5} \cdot 16}$

$\sqrt[5]{16^{5} \cdot 16}$

Step 7

Pull terms out from under the radical.

$16 \sqrt[5]{16}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$16 \sqrt[5]{16}$

Decimal Form:

$27.85761802 \dots$

Related Questions