Simplify fourth root of 81/(4x^2)

$\sqrt[4]{\frac{81}{4 x^{2}}}$

Step 1

Rewrite $81$ as $9^{2}$ .

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

Step 2

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

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

Step 3

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

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

Step 4

Rewrite $\sqrt[4]{{(\frac{9}{2 x})}^{2}}$ as $\sqrt{\sqrt{{(\frac{9}{2 x})}^{2}}}$ .

$\sqrt{\sqrt{{(\frac{9}{2 x})}^{2}}}$

Step 5

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

$\sqrt{\frac{9}{2 x}}$

Step 6

Rewrite $\sqrt{\frac{9}{2 x}}$ as $\frac{\sqrt{9}}{\sqrt{2 x}}$ .

$\frac{\sqrt{9}}{\sqrt{2 x}}$

Step 7

Simplify the numerator.

Rewrite $9$ as $3^{2}$ .

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

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

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

Step 8

Multiply $\frac{3}{\sqrt{2 x}}$ by $\frac{\sqrt{2 x}}{\sqrt{2 x}}$ .

$\frac{3}{\sqrt{2 x}} \cdot \frac{\sqrt{2 x}}{\sqrt{2 x}}$

Step 9

Combine and simplify the denominator.

Multiply $\frac{3}{\sqrt{2 x}}$ by $\frac{\sqrt{2 x}}{\sqrt{2 x}}$ .

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

Raise $\sqrt{2 x}$ to the power of $1$ .

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

Raise $\sqrt{2 x}$ to the power of $1$ .

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify fourth root of 81/(4x^2)

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