Simplify square root of cube root of 8000

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

Step 1

Rewrite $8000$ as $20^{3}$ .

$\sqrt{\sqrt[3]{20^{3}}}$

Step 2

Pull terms out from under the radical, assuming real numbers.

$\sqrt{20}$

Step 3

Rewrite $20$ as $2^{2} \cdot 5$ .

Factor $4$ out of $20$ .

$\sqrt{4 (5)}$

Rewrite $4$ as $2^{2}$ .

$\sqrt{2^{2} \cdot 5}$

Step 4

Pull terms out from under the radical.

$2 \sqrt{5}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$2 \sqrt{5}$

Decimal Form:

$4.47213595 \dots$