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

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

Simplify the numerator.

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

$\frac{\sqrt{216}}{\sqrt[4]{6}}$

Rewrite $216$ as $6^{2} \cdot 6$ .

Factor $36$ out of $216$ .

$\frac{\sqrt{36 (6)}}{\sqrt[4]{6}}$

Rewrite $36$ as $6^{2}$ .

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

Pull terms out from under the radical.

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

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

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

Combine and simplify the denominator.

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

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

Raise $\sqrt[4]{6}$ to the power of $1$ .

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

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

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

Add $1$ and $3$ .

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

Rewrite ${\sqrt[4]{6}}^{4}$ as $6$ .

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

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

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

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

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

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

Cancel the common factor of $4$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Cancel the common factor of $6$ .

Cancel the common factor.

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

Divide $\sqrt{6} {\sqrt[4]{6}}^{3}$ by $1$ .

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

Rewrite ${\sqrt[4]{6}}^{3}$ as ${(6^{3})}^{\frac{1}{4}}$ .

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

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

$\sqrt{6} \sqrt[4]{216}$

Multiply $\sqrt{6} \sqrt[4]{216}$ .

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

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$6^{\frac{1}{2}} \sqrt[4]{216}$

Rewrite $6^{\frac{1}{2}}$ as $6^{\frac{2}{4}}$ .

$6^{\frac{2}{4}} \sqrt[4]{216}$

Rewrite $6^{\frac{2}{4}}$ as $\sqrt[4]{6^{2}}$ .

$\sqrt[4]{6^{2}} \sqrt[4]{216}$

Combine using the product rule for radicals.

$\sqrt[4]{6^{2} \cdot 216}$

Rewrite $216$ as $6^{3}$ .

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

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

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

Add $2$ and $3$ .

$\sqrt[4]{6^{5}}$

Raise $6$ to the power of $5$ .

$\sqrt[4]{7776}$

Rewrite $7776$ as $6^{4} \cdot 6$ .

Factor $1296$ out of $7776$ .

$\sqrt[4]{1296 (6)}$

Rewrite $1296$ as $6^{4}$ .

$\sqrt[4]{6^{4} \cdot 6}$

Pull terms out from under the radical.

$6 \sqrt[4]{6}$

The result can be shown in multiple forms.

Exact Form:

$6 \sqrt[4]{6}$

Decimal Form:

$9.39050748 \dots$

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

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