Simplify ( fourth root of 3)/( fourth root of 6)

$\frac{\sqrt[4]{3}}{\sqrt[4]{6}}$

Step 1

Combine $\sqrt[4]{3}$ and $\sqrt[4]{6}$ into a single radical.

$\sqrt[4]{\frac{3}{6}}$

Step 2

Cancel the common factor of $3$ and $6$ .

Factor $3$ out of $3$ .

$\sqrt[4]{\frac{3 (1)}{6}}$

Cancel the common factors.

Factor $3$ out of $6$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 3

Rewrite $\sqrt[4]{\frac{1}{2}}$ as $\frac{\sqrt[4]{1}}{\sqrt[4]{2}}$ .

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

Step 4

Any root of $1$ is $1$ .

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

Step 5

Multiply $\frac{1}{\sqrt[4]{2}}$ by $\frac{{\sqrt[4]{2}}^{3}}{{\sqrt[4]{2}}^{3}}$ .

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

Step 6

Combine and simplify the denominator.

Multiply $\frac{1}{\sqrt[4]{2}}$ by $\frac{{\sqrt[4]{2}}^{3}}{{\sqrt[4]{2}}^{3}}$ .

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

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

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

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

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

Add $1$ and $3$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $4$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Step 7

Simplify the numerator.

Rewrite ${\sqrt[4]{2}}^{3}$ as $\sqrt[4]{2^{3}}$ .

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

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

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

Step 8

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.84089641 \dots$

Simplify ( fourth root of 3)/( fourth root of 6)

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