Simplify (16 root of 4)/( eighth root of 4)

$\frac{\sqrt[16]{4}}{\sqrt[8]{4}}$

Step 1

Simplify the numerator.

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

$\frac{\sqrt[16]{2^{2}}}{\sqrt[8]{4}}$

Rewrite $\sqrt[16]{2^{2}}$ as $\sqrt[8]{\sqrt{2^{2}}}$ .

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

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

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

Step 2

Simplify the denominator.

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

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

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

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

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

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

Step 3

Multiply $\frac{\sqrt[8]{2}}{\sqrt[4]{2}}$ by $\frac{{\sqrt[4]{2}}^{3}}{{\sqrt[4]{2}}^{3}}$ .

$\frac{\sqrt[8]{2}}{\sqrt[4]{2}} \cdot \frac{{\sqrt[4]{2}}^{3}}{{\sqrt[4]{2}}^{3}}$

Step 4

Combine and simplify the denominator.

Multiply $\frac{\sqrt[8]{2}}{\sqrt[4]{2}}$ by $\frac{{\sqrt[4]{2}}^{3}}{{\sqrt[4]{2}}^{3}}$ .

$\frac{\sqrt[8]{2} {\sqrt[4]{2}}^{3}}{\sqrt[4]{2} {\sqrt[4]{2}}^{3}}$

Raise $\sqrt[4]{2}$ to the power of $1$ .

$\frac{\sqrt[8]{2} {\sqrt[4]{2}}^{3}}{{\sqrt[4]{2}}^{1} {\sqrt[4]{2}}^{3}}$

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

$\frac{\sqrt[8]{2} {\sqrt[4]{2}}^{3}}{{\sqrt[4]{2}}^{1 + 3}}$

Add $1$ and $3$ .

$\frac{\sqrt[8]{2} {\sqrt[4]{2}}^{3}}{{\sqrt[4]{2}}^{4}}$

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

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{2}$ as $2^{\frac{1}{4}}$ .

$\frac{\sqrt[8]{2} {\sqrt[4]{2}}^{3}}{{(2^{\frac{1}{4}})}^{4}}$

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

$\frac{\sqrt[8]{2} {\sqrt[4]{2}}^{3}}{2^{\frac{1}{4} \cdot 4}}$

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

$\frac{\sqrt[8]{2} {\sqrt[4]{2}}^{3}}{2^{\frac{4}{4}}}$

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{\sqrt[8]{2} {\sqrt[4]{2}}^{3}}{2^{\frac{4}{4}}}$

Rewrite the expression.

$\frac{\sqrt[8]{2} {\sqrt[4]{2}}^{3}}{2^{1}}$

Evaluate the exponent.

$\frac{\sqrt[8]{2} {\sqrt[4]{2}}^{3}}{2}$

Step 5

Simplify the numerator.

Rewrite ${\sqrt[4]{2}}^{3}$ as ${(2^{3})}^{\frac{1}{4}}$ .

$\frac{\sqrt[8]{2} \sqrt[4]{2^{3}}}{2}$

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

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

Step 6

Simplify the numerator.

Rewrite the expression using the least common index of $8$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{8}$ as $8^{\frac{1}{4}}$ .

$\frac{\sqrt[8]{2} \cdot 8^{\frac{1}{4}}}{2}$

Rewrite $8^{\frac{1}{4}}$ as $8^{\frac{2}{8}}$ .

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

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

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

Combine using the product rule for radicals.

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

Rewrite $8$ as $2^{3}$ .

$\frac{\sqrt[8]{2 \cdot {(2^{3})}^{2}}}{2}$

Multiply the exponents in ${(2^{3})}^{2}$ .

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

$\frac{\sqrt[8]{2 \cdot 2^{3 \cdot 2}}}{2}$

Multiply $3$ by $2$ .

$\frac{\sqrt[8]{2 \cdot 2^{6}}}{2}$

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

$\frac{\sqrt[8]{2^{1 + 6}}}{2}$

Add $1$ and $6$ .

$\frac{\sqrt[8]{2^{7}}}{2}$

Step 7

Raise $2$ to the power of $7$ .

$\frac{\sqrt[8]{128}}{2}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt[8]{128}}{2}$

Decimal Form:

$0.91700404 \dots$

Simplify (16 root of 4)/( eighth root of 4)

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