Simplify the Radical Expression fourth root of 16/(x^14)

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

Step 1

Rewrite $\frac{16}{x^{14}}$ as ${(\frac{2}{x^{3}})}^{4} \frac{1}{x^{2}}$ .

Factor the perfect power $2^{4}$ out of $16$ .

$\sqrt[4]{\frac{2^{4} \cdot 1}{x^{14}}}$

Factor the perfect power ${(x^{3})}^{4}$ out of $x^{14}$ .

$\sqrt[4]{\frac{2^{4} \cdot 1}{{(x^{3})}^{4} x^{2}}}$

Rearrange the fraction $\frac{2^{4} \cdot 1}{{(x^{3})}^{4} x^{2}}$ .

$\sqrt[4]{{(\frac{2}{x^{3}})}^{4} \frac{1}{x^{2}}}$

Step 2

Pull terms out from under the radical.

$| \frac{2}{x^{3}} | \sqrt[4]{\frac{1}{x^{2}}}$

Step 3

Remove non-negative terms from the absolute value.

$\frac{2}{| x^{3} |} \sqrt[4]{\frac{1}{x^{2}}}$

Step 4

Rewrite $1$ as $1^{2}$ .

$\frac{2}{| x^{3} |} \sqrt[4]{\frac{1^{2}}{x^{2}}}$

Step 5

Rewrite $\frac{1^{2}}{x^{2}}$ as ${(\frac{1}{x})}^{2}$ .

$\frac{2}{| x^{3} |} \sqrt[4]{{(\frac{1}{x})}^{2}}$

Step 6

Rewrite $\sqrt[4]{{(\frac{1}{x})}^{2}}$ as $\sqrt{\sqrt{{(\frac{1}{x})}^{2}}}$ .

$\frac{2}{| x^{3} |} \sqrt{\sqrt{{(\frac{1}{x})}^{2}}}$

Step 7

Pull terms out from under the radical.

$\frac{2}{| x^{3} |} \sqrt{| \frac{1}{x} |}$

Step 8

Combine $\frac{2}{| x^{3} |}$ and $\sqrt{| \frac{1}{x} |}$ .

$\frac{2 \sqrt{| \frac{1}{x} |}}{| x^{3} |}$