Simplify fifth root of fourth root of cube root of square root of 8

$\sqrt[5]{\sqrt[4]{\sqrt[3]{\sqrt{8}}}}$

Step 1

Rewrite $8$ as $2^{2} \cdot 2$ .

Factor $4$ out of $8$ .

$\sqrt[5]{\sqrt[4]{\sqrt[3]{\sqrt{4 (2)}}}}$

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

$\sqrt[5]{\sqrt[4]{\sqrt[3]{\sqrt{2^{2} \cdot 2}}}}$

Step 2

Pull terms out from under the radical.

$\sqrt[5]{\sqrt[4]{\sqrt[3]{2 \sqrt{2}}}}$

Step 3

Rewrite $\sqrt[4]{\sqrt[3]{2 \sqrt{2}}}$ as $\sqrt[12]{2 \sqrt{2}}$ .

$\sqrt[5]{\sqrt[12]{2 \sqrt{2}}}$

Step 4

Rewrite $\sqrt[5]{\sqrt[12]{2 \sqrt{2}}}$ as $\sqrt[60]{2 \sqrt{2}}$ .

$\sqrt[60]{2 \sqrt{2}}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\sqrt[60]{2 \sqrt{2}}$

Decimal Form:

$1.01747969 \dots$