Simplify fourth root of 16x^8* fourth root of x^14

$\sqrt[4]{16 x^{8}} \cdot \sqrt[4]{x^{14}}$

Step 1

Rewrite $16 x^{8}$ as ${(2 x^{2})}^{4}$ .

$\sqrt[4]{{(2 x^{2})}^{4}} \cdot \sqrt[4]{x^{14}}$

Step 2

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

$2 x^{2} \cdot \sqrt[4]{x^{14}}$

Step 3

Rewrite $x^{14}$ as ${(x^{3})}^{4} x^{2}$ .

Factor out $x^{12}$ .

$2 x^{2} \cdot \sqrt[4]{x^{12} x^{2}}$

Rewrite $x^{12}$ as ${(x^{3})}^{4}$ .

$2 x^{2} \cdot \sqrt[4]{{(x^{3})}^{4} x^{2}}$

Step 4

Pull terms out from under the radical.

$2 x^{2} \cdot (x^{3} \sqrt[4]{x^{2}})$

Step 5

Multiply $x^{2}$ by $x^{3}$ by adding the exponents.

Move $x^{3}$ .

$2 (x^{3} x^{2}) \cdot \sqrt[4]{x^{2}}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Add $3$ and $2$ .

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

Step 6

Rewrite $\sqrt[4]{x^{2}}$ as $\sqrt{\sqrt{x^{2}}}$ .

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

Step 7

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

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