Evaluate fourth root of 16/27

$\sqrt[4]{\frac{16}{27}}$

Step 1

Rewrite $\sqrt[4]{\frac{16}{27}}$ as $\frac{\sqrt[4]{16}}{\sqrt[4]{27}}$ .

$\frac{\sqrt[4]{16}}{\sqrt[4]{27}}$

Step 2

Simplify the numerator.

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

$\frac{\sqrt[4]{2^{4}}}{\sqrt[4]{27}}$

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

$\frac{2}{\sqrt[4]{27}}$

Step 3

Multiply $\frac{2}{\sqrt[4]{27}}$ by $\frac{{\sqrt[4]{27}}^{3}}{{\sqrt[4]{27}}^{3}}$ .

$\frac{2}{\sqrt[4]{27}} \cdot \frac{{\sqrt[4]{27}}^{3}}{{\sqrt[4]{27}}^{3}}$

Step 4

Combine and simplify the denominator.

Multiply $\frac{2}{\sqrt[4]{27}}$ by $\frac{{\sqrt[4]{27}}^{3}}{{\sqrt[4]{27}}^{3}}$ .

$\frac{2 {\sqrt[4]{27}}^{3}}{\sqrt[4]{27} {\sqrt[4]{27}}^{3}}$

Raise $\sqrt[4]{27}$ to the power of $1$ .

$\frac{2 {\sqrt[4]{27}}^{3}}{{\sqrt[4]{27}}^{1} {\sqrt[4]{27}}^{3}}$

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

$\frac{2 {\sqrt[4]{27}}^{3}}{{\sqrt[4]{27}}^{1 + 3}}$

Add $1$ and $3$ .

$\frac{2 {\sqrt[4]{27}}^{3}}{{\sqrt[4]{27}}^{4}}$

Rewrite ${\sqrt[4]{27}}^{4}$ as $27$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{27}$ as $27^{\frac{1}{4}}$ .

$\frac{2 {\sqrt[4]{27}}^{3}}{{(27^{\frac{1}{4}})}^{4}}$

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

$\frac{2 {\sqrt[4]{27}}^{3}}{27^{\frac{1}{4} \cdot 4}}$

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

$\frac{2 {\sqrt[4]{27}}^{3}}{27^{\frac{4}{4}}}$

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{2 {\sqrt[4]{27}}^{3}}{27^{\frac{4}{4}}}$

Rewrite the expression.

$\frac{2 {\sqrt[4]{27}}^{3}}{27^{1}}$

Evaluate the exponent.

$\frac{2 {\sqrt[4]{27}}^{3}}{27}$

Step 5

Simplify the numerator.

Rewrite ${\sqrt[4]{27}}^{3}$ as $\sqrt[4]{27^{3}}$ .

$\frac{2 \sqrt[4]{27^{3}}}{27}$

Raise $27$ to the power of $3$ .

$\frac{2 \sqrt[4]{19683}}{27}$

Rewrite $19683$ as $9^{4} \cdot 3$ .

Factor $6561$ out of $19683$ .

$\frac{2 \sqrt[4]{6561 (3)}}{27}$

Rewrite $6561$ as $9^{4}$ .

$\frac{2 \sqrt[4]{9^{4} \cdot 3}}{27}$

Pull terms out from under the radical.

$\frac{2 \cdot 9 \sqrt[4]{3}}{27}$

Multiply $2$ by $9$ .

$\frac{18 \sqrt[4]{3}}{27}$

Step 6

Cancel the common factor of $18$ and $27$ .

Factor $9$ out of $18 \sqrt[4]{3}$ .

$\frac{9 (2 \sqrt[4]{3})}{27}$

Cancel the common factors.

Factor $9$ out of $27$ .

$\frac{9 (2 \sqrt[4]{3})}{9 (3)}$

Cancel the common factor.

$\frac{9 (2 \sqrt[4]{3})}{9 \cdot 3}$

Rewrite the expression.

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

Step 7

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.87738267 \dots$

Evaluate fourth root of 16/27

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