Evaluate fourth root of 256/125

$\sqrt[4]{\frac{256}{125}}$

Step 1

Rewrite $\sqrt[4]{\frac{256}{125}}$ as $\frac{\sqrt[4]{256}}{\sqrt[4]{125}}$ .

$\frac{\sqrt[4]{256}}{\sqrt[4]{125}}$

Step 2

Simplify the numerator.

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

$\frac{\sqrt[4]{4^{4}}}{\sqrt[4]{125}}$

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

$\frac{4}{\sqrt[4]{125}}$

Step 3

Multiply $\frac{4}{\sqrt[4]{125}}$ by $\frac{{\sqrt[4]{125}}^{3}}{{\sqrt[4]{125}}^{3}}$ .

$\frac{4}{\sqrt[4]{125}} \cdot \frac{{\sqrt[4]{125}}^{3}}{{\sqrt[4]{125}}^{3}}$

Step 4

Combine and simplify the denominator.

Multiply $\frac{4}{\sqrt[4]{125}}$ by $\frac{{\sqrt[4]{125}}^{3}}{{\sqrt[4]{125}}^{3}}$ .

$\frac{4 {\sqrt[4]{125}}^{3}}{\sqrt[4]{125} {\sqrt[4]{125}}^{3}}$

Raise $\sqrt[4]{125}$ to the power of $1$ .

$\frac{4 {\sqrt[4]{125}}^{3}}{{\sqrt[4]{125}}^{1} {\sqrt[4]{125}}^{3}}$

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

$\frac{4 {\sqrt[4]{125}}^{3}}{{\sqrt[4]{125}}^{1 + 3}}$

Add $1$ and $3$ .

$\frac{4 {\sqrt[4]{125}}^{3}}{{\sqrt[4]{125}}^{4}}$

Rewrite ${\sqrt[4]{125}}^{4}$ as $125$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{125}$ as $125^{\frac{1}{4}}$ .

$\frac{4 {\sqrt[4]{125}}^{3}}{{(125^{\frac{1}{4}})}^{4}}$

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

$\frac{4 {\sqrt[4]{125}}^{3}}{125^{\frac{1}{4} \cdot 4}}$

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

$\frac{4 {\sqrt[4]{125}}^{3}}{125^{\frac{4}{4}}}$

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 {\sqrt[4]{125}}^{3}}{125^{\frac{4}{4}}}$

Rewrite the expression.

$\frac{4 {\sqrt[4]{125}}^{3}}{125^{1}}$

Evaluate the exponent.

$\frac{4 {\sqrt[4]{125}}^{3}}{125}$

Step 5

Simplify the numerator.

Rewrite ${\sqrt[4]{125}}^{3}$ as $\sqrt[4]{125^{3}}$ .

$\frac{4 \sqrt[4]{125^{3}}}{125}$

Raise $125$ to the power of $3$ .

$\frac{4 \sqrt[4]{1953125}}{125}$

Rewrite $1953125$ as $25^{4} \cdot 5$ .

Factor $390625$ out of $1953125$ .

$\frac{4 \sqrt[4]{390625 (5)}}{125}$

Rewrite $390625$ as $25^{4}$ .

$\frac{4 \sqrt[4]{25^{4} \cdot 5}}{125}$

Pull terms out from under the radical.

$\frac{4 \cdot 25 \sqrt[4]{5}}{125}$

Multiply $4$ by $25$ .

$\frac{100 \sqrt[4]{5}}{125}$

Step 6

Cancel the common factor of $100$ and $125$ .

Factor $25$ out of $100 \sqrt[4]{5}$ .

$\frac{25 (4 \sqrt[4]{5})}{125}$

Cancel the common factors.

Factor $25$ out of $125$ .

$\frac{25 (4 \sqrt[4]{5})}{25 (5)}$

Cancel the common factor.

$\frac{25 (4 \sqrt[4]{5})}{25 \cdot 5}$

Rewrite the expression.

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

Step 7

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$1.19627902 \dots$

Evaluate fourth root of 256/125

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