Simplify the Radical Expression fourth root of 5/8

$\sqrt[4]{\frac{5}{8}}$

Step 1

Rewrite $\sqrt[4]{\frac{5}{8}}$ as $\frac{\sqrt[4]{5}}{\sqrt[4]{8}}$ .

$\frac{\sqrt[4]{5}}{\sqrt[4]{8}}$

Step 2

Multiply $\frac{\sqrt[4]{5}}{\sqrt[4]{8}}$ by $\frac{{\sqrt[4]{8}}^{3}}{{\sqrt[4]{8}}^{3}}$ .

$\frac{\sqrt[4]{5}}{\sqrt[4]{8}} \cdot \frac{{\sqrt[4]{8}}^{3}}{{\sqrt[4]{8}}^{3}}$

Step 3

Combine and simplify the denominator.

Multiply $\frac{\sqrt[4]{5}}{\sqrt[4]{8}}$ by $\frac{{\sqrt[4]{8}}^{3}}{{\sqrt[4]{8}}^{3}}$ .

$\frac{\sqrt[4]{5} {\sqrt[4]{8}}^{3}}{\sqrt[4]{8} {\sqrt[4]{8}}^{3}}$

Raise $\sqrt[4]{8}$ to the power of $1$ .

$\frac{\sqrt[4]{5} {\sqrt[4]{8}}^{3}}{{\sqrt[4]{8}}^{1} {\sqrt[4]{8}}^{3}}$

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

$\frac{\sqrt[4]{5} {\sqrt[4]{8}}^{3}}{{\sqrt[4]{8}}^{1 + 3}}$

Add $1$ and $3$ .

$\frac{\sqrt[4]{5} {\sqrt[4]{8}}^{3}}{{\sqrt[4]{8}}^{4}}$

Rewrite ${\sqrt[4]{8}}^{4}$ as $8$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{8}$ as $8^{\frac{1}{4}}$ .

$\frac{\sqrt[4]{5} {\sqrt[4]{8}}^{3}}{{(8^{\frac{1}{4}})}^{4}}$

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

$\frac{\sqrt[4]{5} {\sqrt[4]{8}}^{3}}{8^{\frac{1}{4} \cdot 4}}$

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

$\frac{\sqrt[4]{5} {\sqrt[4]{8}}^{3}}{8^{\frac{4}{4}}}$

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{\sqrt[4]{5} {\sqrt[4]{8}}^{3}}{8^{\frac{4}{4}}}$

Rewrite the expression.

$\frac{\sqrt[4]{5} {\sqrt[4]{8}}^{3}}{8^{1}}$

Evaluate the exponent.

$\frac{\sqrt[4]{5} {\sqrt[4]{8}}^{3}}{8}$

Step 4

Simplify the numerator.

Rewrite ${\sqrt[4]{8}}^{3}$ as ${(8^{3})}^{\frac{1}{4}}$ .

$\frac{\sqrt[4]{5} \sqrt[4]{8^{3}}}{8}$

Raise $8$ to the power of $3$ .

$\frac{\sqrt[4]{5} \sqrt[4]{512}}{8}$

Rewrite $512$ as $4^{4} \cdot 2$ .

Factor $256$ out of $512$ .

$\frac{\sqrt[4]{5} \sqrt[4]{256 (2)}}{8}$

Rewrite $256$ as $4^{4}$ .

$\frac{\sqrt[4]{5} \sqrt[4]{4^{4} \cdot 2}}{8}$

Pull terms out from under the radical.

$\frac{\sqrt[4]{5} (| 4 | \sqrt[4]{2})}{8}$

The absolute value is the distance between a number and zero. The distance between $0$ and $4$ is $4$ .

$\frac{\sqrt[4]{5} \cdot 4 \sqrt[4]{2}}{8}$

Combine exponents.

Combine using the product rule for radicals.

$\frac{4 \sqrt[4]{5 \cdot 2}}{8}$

Multiply $5$ by $2$ .

$\frac{4 \sqrt[4]{10}}{8}$

Step 5

Cancel the common factor of $4$ and $8$ .

Factor $4$ out of $4 \sqrt[4]{10}$ .

$\frac{4 (\sqrt[4]{10})}{8}$

Cancel the common factors.

Factor $4$ out of $8$ .

$\frac{4 \sqrt[4]{10}}{4 \cdot 2}$

Cancel the common factor.

$\frac{4 \sqrt[4]{10}}{4 \cdot 2}$

Rewrite the expression.

$\frac{\sqrt[4]{10}}{2}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt[4]{10}}{2}$

Decimal Form:

$0.88913970 \dots$

Simplify the Radical Expression fourth root of 5/8

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