Simplify cube root of square root of 8000

$\sqrt[3]{\sqrt{8000}}$

Step 1

Rewrite $8000$ as $40^{2} \cdot 5$ .

Factor $1600$ out of $8000$ .

$\sqrt[3]{\sqrt{1600 (5)}}$

Rewrite $1600$ as $40^{2}$ .

$\sqrt[3]{\sqrt{40^{2} \cdot 5}}$

Step 2

Pull terms out from under the radical.

$\sqrt[3]{40 \sqrt{5}}$

Step 3

Rewrite $40 \sqrt{5}$ as $2^{3} \cdot (5 \sqrt{5})$ .

Factor $8$ out of $40$ .

$\sqrt[3]{8 (5) \sqrt{5}}$

Rewrite $8$ as $2^{3}$ .

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

Add parentheses.

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

Step 4

Pull terms out from under the radical.

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

Step 5

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$4.47213595 \dots$