Algebra Examples

$\sqrt[3]{\sqrt[5]{- 64}}$

Step 1

Rewrite $- 64$ as ${(- 2)}^{5} \cdot 2$ .

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Step 1.1

Factor $- 32$ out of $- 64$ .

$\sqrt[3]{\sqrt[5]{- 32 (2)}}$

Step 1.2

Rewrite $- 32$ as ${(- 2)}^{5}$ .

$\sqrt[3]{\sqrt[5]{{(- 2)}^{5} \cdot 2}}$

Step 2

Pull terms out from under the radical.

$\sqrt[3]{- 2 \sqrt[5]{2}}$

Step 3

Rewrite $- 2 \sqrt[5]{2}$ as ${({(- 1)}^{3})}^{3} \cdot (2 \sqrt[5]{2})$ .

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Step 3.1

Rewrite $- 2$ as $- 1 (2)$ .

$\sqrt[3]{- 1 (2) \sqrt[5]{2}}$

Step 3.2

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

$\sqrt[3]{{(- 1)}^{3} \cdot 2 \sqrt[5]{2}}$

Step 3.3

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

$\sqrt[3]{{({(- 1)}^{3})}^{3} \cdot 2 \sqrt[5]{2}}$

Step 3.4

Add parentheses.

$\sqrt[3]{{({(- 1)}^{3})}^{3} \cdot (2 \sqrt[5]{2})}$

Step 4

Pull terms out from under the radical.

${(- 1)}^{3} \sqrt[3]{2 \sqrt[5]{2}}$

Step 5

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

$- 1 \sqrt[3]{2 \sqrt[5]{2}}$

Step 6

Rewrite $- 1 \sqrt[3]{2 \sqrt[5]{2}}$ as $- \sqrt[3]{2 \sqrt[5]{2}}$ .

$- \sqrt[3]{2 \sqrt[5]{2}}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$- \sqrt[3]{2 \sqrt[5]{2}}$

Decimal Form:

$- 1.31950791 \dots$