Simplify ( fourth root of 6)/( cube root of 2)

$\frac{\sqrt[4]{6}}{\sqrt[3]{2}}$

Step 1

Multiply $\frac{\sqrt[4]{6}}{\sqrt[3]{2}}$ by $\frac{{\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{2}}$ .

$\frac{\sqrt[4]{6}}{\sqrt[3]{2}} \cdot \frac{{\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{2}}$

Step 2

Combine and simplify the denominator.

Multiply $\frac{\sqrt[4]{6}}{\sqrt[3]{2}}$ by $\frac{{\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{2}}$ .

$\frac{\sqrt[4]{6} {\sqrt[3]{2}}^{2}}{\sqrt[3]{2} {\sqrt[3]{2}}^{2}}$

Raise $\sqrt[3]{2}$ to the power of $1$ .

$\frac{\sqrt[4]{6} {\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{1} {\sqrt[3]{2}}^{2}}$

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

$\frac{\sqrt[4]{6} {\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{1 + 2}}$

Add $1$ and $2$ .

$\frac{\sqrt[4]{6} {\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{3}}$

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

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{2}$ as $2^{\frac{1}{3}}$ .

$\frac{\sqrt[4]{6} {\sqrt[3]{2}}^{2}}{{(2^{\frac{1}{3}})}^{3}}$

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

$\frac{\sqrt[4]{6} {\sqrt[3]{2}}^{2}}{2^{\frac{1}{3} \cdot 3}}$

Combine $\frac{1}{3}$ and $3$ .

$\frac{\sqrt[4]{6} {\sqrt[3]{2}}^{2}}{2^{\frac{3}{3}}}$

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{\sqrt[4]{6} {\sqrt[3]{2}}^{2}}{2^{\frac{3}{3}}}$

Rewrite the expression.

$\frac{\sqrt[4]{6} {\sqrt[3]{2}}^{2}}{2^{1}}$

Evaluate the exponent.

$\frac{\sqrt[4]{6} {\sqrt[3]{2}}^{2}}{2}$

Step 3

Simplify the numerator.

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

$\frac{\sqrt[4]{6} \sqrt[3]{2^{2}}}{2}$

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

$\frac{\sqrt[4]{6} \sqrt[3]{4}}{2}$

Step 4

Simplify the numerator.

Rewrite the expression using the least common index of $12$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{6}$ as $6^{\frac{1}{4}}$ .

$\frac{6^{\frac{1}{4}} \sqrt[3]{4}}{2}$

Rewrite $6^{\frac{1}{4}}$ as $6^{\frac{3}{12}}$ .

$\frac{6^{\frac{3}{12}} \sqrt[3]{4}}{2}$

Rewrite $6^{\frac{3}{12}}$ as $\sqrt[12]{6^{3}}$ .

$\frac{\sqrt[12]{6^{3}} \sqrt[3]{4}}{2}$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{4}$ as $4^{\frac{1}{3}}$ .

$\frac{\sqrt[12]{6^{3}} \cdot 4^{\frac{1}{3}}}{2}$

Rewrite $4^{\frac{1}{3}}$ as $4^{\frac{4}{12}}$ .

$\frac{\sqrt[12]{6^{3}} \cdot 4^{\frac{4}{12}}}{2}$

Rewrite $4^{\frac{4}{12}}$ as $\sqrt[12]{4^{4}}$ .

$\frac{\sqrt[12]{6^{3}} \sqrt[12]{4^{4}}}{2}$

Combine using the product rule for radicals.

$\frac{\sqrt[12]{6^{3} \cdot 4^{4}}}{2}$

Raise $6$ to the power of $3$ .

$\frac{\sqrt[12]{216 \cdot 4^{4}}}{2}$

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

$\frac{\sqrt[12]{216 \cdot 256}}{2}$

Step 5

Multiply $216$ by $256$ .

$\frac{\sqrt[12]{55296}}{2}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt[12]{55296}}{2}$

Decimal Form:

$1.24220845 \dots$

Simplify ( fourth root of 6)/( cube root of 2)

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