Simplify the Radical Expression fifth root of 27/64

$\sqrt[5]{\frac{27}{64}}$

Step 1

Rewrite $\sqrt[5]{\frac{27}{64}}$ as $\frac{\sqrt[5]{27}}{\sqrt[5]{64}}$ .

$\frac{\sqrt[5]{27}}{\sqrt[5]{64}}$

Step 2

Simplify the denominator.

Rewrite $64$ as $2^{5} \cdot 2$ .

Factor $32$ out of $64$ .

$\frac{\sqrt[5]{27}}{\sqrt[5]{32 (2)}}$

Rewrite $32$ as $2^{5}$ .

$\frac{\sqrt[5]{27}}{\sqrt[5]{2^{5} \cdot 2}}$

Pull terms out from under the radical.

$\frac{\sqrt[5]{27}}{2 \sqrt[5]{2}}$

Step 3

Multiply $\frac{\sqrt[5]{27}}{2 \sqrt[5]{2}}$ by $\frac{{\sqrt[5]{2}}^{4}}{{\sqrt[5]{2}}^{4}}$ .

$\frac{\sqrt[5]{27}}{2 \sqrt[5]{2}} \cdot \frac{{\sqrt[5]{2}}^{4}}{{\sqrt[5]{2}}^{4}}$

Step 4

Combine and simplify the denominator.

Multiply $\frac{\sqrt[5]{27}}{2 \sqrt[5]{2}}$ by $\frac{{\sqrt[5]{2}}^{4}}{{\sqrt[5]{2}}^{4}}$ .

$\frac{\sqrt[5]{27} {\sqrt[5]{2}}^{4}}{2 \sqrt[5]{2} {\sqrt[5]{2}}^{4}}$

Move $\sqrt[5]{2}$ .

$\frac{\sqrt[5]{27} {\sqrt[5]{2}}^{4}}{2 (\sqrt[5]{2} {\sqrt[5]{2}}^{4})}$

Raise $\sqrt[5]{2}$ to the power of $1$ .

$\frac{\sqrt[5]{27} {\sqrt[5]{2}}^{4}}{2 ({\sqrt[5]{2}}^{1} {\sqrt[5]{2}}^{4})}$

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

$\frac{\sqrt[5]{27} {\sqrt[5]{2}}^{4}}{2 {\sqrt[5]{2}}^{1 + 4}}$

Add $1$ and $4$ .

$\frac{\sqrt[5]{27} {\sqrt[5]{2}}^{4}}{2 {\sqrt[5]{2}}^{5}}$

Rewrite ${\sqrt[5]{2}}^{5}$ as $2$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{2}$ as $2^{\frac{1}{5}}$ .

$\frac{\sqrt[5]{27} {\sqrt[5]{2}}^{4}}{2 {(2^{\frac{1}{5}})}^{5}}$

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

$\frac{\sqrt[5]{27} {\sqrt[5]{2}}^{4}}{2 \cdot 2^{\frac{1}{5} \cdot 5}}$

Combine $\frac{1}{5}$ and $5$ .

$\frac{\sqrt[5]{27} {\sqrt[5]{2}}^{4}}{2 \cdot 2^{\frac{5}{5}}}$

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{\sqrt[5]{27} {\sqrt[5]{2}}^{4}}{2 \cdot 2^{\frac{5}{5}}}$

Rewrite the expression.

$\frac{\sqrt[5]{27} {\sqrt[5]{2}}^{4}}{2 \cdot 2^{1}}$

Evaluate the exponent.

$\frac{\sqrt[5]{27} {\sqrt[5]{2}}^{4}}{2 \cdot 2}$

Step 5

Simplify the numerator.

Rewrite ${\sqrt[5]{2}}^{4}$ as ${(2^{4})}^{\frac{1}{5}}$ .

$\frac{\sqrt[5]{27} \sqrt[5]{2^{4}}}{2 \cdot 2}$

Raise $2$ to the power of $4$ .

$\frac{\sqrt[5]{27} \sqrt[5]{16}}{2 \cdot 2}$

Step 6

Multiply $2$ by $2$ .

$\frac{\sqrt[5]{27} \sqrt[5]{16}}{4}$

Step 7

Simplify the numerator.

Combine using the product rule for radicals.

$\frac{\sqrt[5]{27 \cdot 16}}{4}$

Multiply $27$ by $16$ .

$\frac{\sqrt[5]{432}}{4}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt[5]{432}}{4}$

Decimal Form:

$0.84146635 \dots$

Simplify the Radical Expression fifth root of 27/64

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