Simplify ( fifth root of -21)^6

${(\sqrt[5]{- 21})}^{6}$

Step 1

Rewrite $- 21$ as ${(- 1)}^{5} \cdot 21$ .

Rewrite $- 21$ as $- 1 (21)$ .

${\sqrt[5]{- 1 (21)}}^{6}$

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

${\sqrt[5]{{(- 1)}^{5} \cdot 21}}^{6}$

${\sqrt[5]{{(- 1)}^{5} \cdot 21}}^{6}$

Step 2

Pull terms out from under the radical.

${(- 1 \sqrt[5]{21})}^{6}$

Step 3

Simplify the expression.

Rewrite $- 1 \sqrt[5]{21}$ as $- \sqrt[5]{21}$ .

${(- \sqrt[5]{21})}^{6}$

Apply the product rule to $- \sqrt[5]{21}$ .

${(- 1)}^{6} {\sqrt[5]{21}}^{6}$

Raise $- 1$ to the power of $6$ .

$1 {\sqrt[5]{21}}^{6}$

Multiply ${\sqrt[5]{21}}^{6}$ by $1$ .

${\sqrt[5]{21}}^{6}$

${\sqrt[5]{21}}^{6}$

Step 4

Rewrite ${\sqrt[5]{21}}^{6}$ as ${(21^{6})}^{\frac{1}{5}}$ .

$\sqrt[5]{21^{6}}$

Step 5

Raise $21$ to the power of $6$ .

$\sqrt[5]{85766121}$

Step 6

Rewrite $85766121$ as $21^{5} \cdot 21$ .

Factor $4084101$ out of $85766121$ .

$\sqrt[5]{4084101 (21)}$

Rewrite $4084101$ as $21^{5}$ .

$\sqrt[5]{21^{5} \cdot 21}$

$\sqrt[5]{21^{5} \cdot 21}$

Step 7

Pull terms out from under the radical.

$21 \sqrt[5]{21}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$21 \sqrt[5]{21}$

Decimal Form:

$38.60674203 \dots$

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