Simplify fourth root of 81/(4x)

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

Step 1

Rewrite $\sqrt[4]{\frac{81}{4 x}}$ as $\frac{\sqrt[4]{81}}{\sqrt[4]{4 x}}$ .

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

Step 2

Simplify the numerator.

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

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

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

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

Step 3

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

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

Step 4

Combine and simplify the denominator.

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

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

Raise $\sqrt[4]{4 x}$ to the power of $1$ .

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

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

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

Add $1$ and $3$ .

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

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

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{4 x}$ as ${(4 x)}^{\frac{1}{4}}$ .

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

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

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

Combine $\frac{1}{4}$ and $4$ .

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

Cancel the common factor of $4$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Step 5

Simplify the numerator.

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

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

Apply the product rule to $4 x$ .

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

Raise $4$ to the power of $3$ .

$\frac{3 \sqrt[4]{64 x^{3}}}{4 x}$

Rewrite $64 x^{3}$ as $2^{4} \cdot (4 x^{3})$ .

Factor $16$ out of $64$ .

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

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

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

Add parentheses.

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

Pull terms out from under the radical.

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

Multiply $3$ by $2$ .

$\frac{6 \sqrt[4]{4 x^{3}}}{4 x}$

Step 6

Cancel the common factor of $6$ and $4$ .

Factor $2$ out of $6 \sqrt[4]{4 x^{3}}$ .

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

Cancel the common factors.

Factor $2$ out of $4 x$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Simplify fourth root of 81/(4x)

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