Simplify fourth root of 256/(9x^2)

$\sqrt[4]{\frac{256}{9 x^{2}}}$

Step 1

Rewrite $256$ as $16^{2}$ .

$\sqrt[4]{\frac{16^{2}}{9 x^{2}}}$

Step 2

Rewrite $9 x^{2}$ as ${(3 x)}^{2}$ .

$\sqrt[4]{\frac{16^{2}}{{(3 x)}^{2}}}$

Step 3

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

$\sqrt[4]{{(\frac{16}{3 x})}^{2}}$

Step 4

Rewrite $\sqrt[4]{{(\frac{16}{3 x})}^{2}}$ as $\sqrt{\sqrt{{(\frac{16}{3 x})}^{2}}}$ .

$\sqrt{\sqrt{{(\frac{16}{3 x})}^{2}}}$

Step 5

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

$\sqrt{\frac{16}{3 x}}$

Step 6

Rewrite $\sqrt{\frac{16}{3 x}}$ as $\frac{\sqrt{16}}{\sqrt{3 x}}$ .

$\frac{\sqrt{16}}{\sqrt{3 x}}$

Step 7

Simplify the numerator.

Rewrite $16$ as $4^{2}$ .

$\frac{\sqrt{4^{2}}}{\sqrt{3 x}}$

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

$\frac{4}{\sqrt{3 x}}$

Step 8

Multiply $\frac{4}{\sqrt{3 x}}$ by $\frac{\sqrt{3 x}}{\sqrt{3 x}}$ .

$\frac{4}{\sqrt{3 x}} \cdot \frac{\sqrt{3 x}}{\sqrt{3 x}}$

Step 9

Combine and simplify the denominator.

Multiply $\frac{4}{\sqrt{3 x}}$ by $\frac{\sqrt{3 x}}{\sqrt{3 x}}$ .

$\frac{4 \sqrt{3 x}}{\sqrt{3 x} \sqrt{3 x}}$

Raise $\sqrt{3 x}$ to the power of $1$ .

$\frac{4 \sqrt{3 x}}{{\sqrt{3 x}}^{1} \sqrt{3 x}}$

Raise $\sqrt{3 x}$ to the power of $1$ .

$\frac{4 \sqrt{3 x}}{{\sqrt{3 x}}^{1} {\sqrt{3 x}}^{1}}$

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

$\frac{4 \sqrt{3 x}}{{\sqrt{3 x}}^{1 + 1}}$

Add $1$ and $1$ .

$\frac{4 \sqrt{3 x}}{{\sqrt{3 x}}^{2}}$

Rewrite ${\sqrt{3 x}}^{2}$ as $3 x$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 x}$ as ${(3 x)}^{\frac{1}{2}}$ .

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

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Combine $\frac{1}{2}$ and $2$ .

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

$\frac{4 \sqrt{3 x}}{3 x}$

Simplify fourth root of 256/(9x^2)

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