Evaluate eighth root of 75^4

$\sqrt[8]{75^{4}}$

Step 1

Raise $75$ to the power of $4$ .

$\sqrt[8]{31640625}$

Step 2

Rewrite $31640625$ as $5^{8} \cdot 81$ .

Factor $390625$ out of $31640625$ .

$\sqrt[8]{390625 (81)}$

Rewrite $390625$ as $5^{8}$ .

$\sqrt[8]{5^{8} \cdot 81}$

Step 3

Pull terms out from under the radical.

$5 \sqrt[8]{81}$

Step 4

Rewrite $81$ as $3^{4}$ .

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

Step 5

Rewrite $\sqrt[8]{3^{4}}$ as $\sqrt{\sqrt[4]{3^{4}}}$ .

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

Step 6

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

$5 \sqrt{3}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$5 \sqrt{3}$

Decimal Form:

$8.66025403 \dots$