Simplify (10 root of 4)/( fifth root of 4)

$\frac{\sqrt[10]{4}}{\sqrt[5]{4}}$

Step 1

Simplify the numerator.

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

$\frac{\sqrt[10]{2^{2}}}{\sqrt[5]{4}}$

Rewrite $\sqrt[10]{2^{2}}$ as $\sqrt[5]{\sqrt{2^{2}}}$ .

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

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

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

Step 2

Combine $\sqrt[5]{2}$ and $\sqrt[5]{4}$ into a single radical.

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

Step 3

Cancel the common factor of $2$ and $4$ .

Factor $2$ out of $2$ .

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

Cancel the common factors.

Factor $2$ out of $4$ .

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

Cancel the common factor.

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

Rewrite the expression.

$\sqrt[5]{\frac{1}{2}}$

Step 4

Rewrite $\sqrt[5]{\frac{1}{2}}$ as $\frac{\sqrt[5]{1}}{\sqrt[5]{2}}$ .

$\frac{\sqrt[5]{1}}{\sqrt[5]{2}}$

Step 5

Any root of $1$ is $1$ .

$\frac{1}{\sqrt[5]{2}}$

Step 6

Multiply $\frac{1}{\sqrt[5]{2}}$ by $\frac{{\sqrt[5]{2}}^{4}}{{\sqrt[5]{2}}^{4}}$ .

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

Step 7

Combine and simplify the denominator.

Multiply $\frac{1}{\sqrt[5]{2}}$ by $\frac{{\sqrt[5]{2}}^{4}}{{\sqrt[5]{2}}^{4}}$ .

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

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

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

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

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

Add $1$ and $4$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $5$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Step 8

Simplify the numerator.

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

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

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

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

Step 9

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.87055056 \dots$

Simplify (10 root of 4)/( fifth root of 4)

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