Simplify ( eighth root of 4)/( fourth root of 4)

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

Step 1

Simplify the numerator.

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

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

Rewrite $\sqrt[8]{2^{2}}$ as $\sqrt[4]{\sqrt{2^{2}}}$ .

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

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

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

Step 2

Simplify the denominator.

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

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

Rewrite $\sqrt[4]{2^{2}}$ as $\sqrt{\sqrt{2^{2}}}$ .

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

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

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

Step 3

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

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

Step 4

Combine and simplify the denominator.

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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

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

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

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

Combine $\frac{1}{2}$ and $2$ .

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Step 5

Simplify the numerator.

Rewrite the expression using the least common index of $4$ .

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

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

Rewrite $2^{\frac{1}{2}}$ as $2^{\frac{2}{4}}$ .

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

Rewrite $2^{\frac{2}{4}}$ as $\sqrt[4]{2^{2}}$ .

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

Combine using the product rule for radicals.

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

Multiply $2$ by $2^{2}$ by adding the exponents.

Multiply $2$ by $2^{2}$ .

Raise $2$ to the power of $1$ .

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

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

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

Add $1$ and $2$ .

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

Step 6

Raise $2$ to the power of $3$ .

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

Step 7

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.84089641 \dots$

Simplify ( eighth root of 4)/( fourth root of 4)

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