Evaluate sixth root of 81/4

$\sqrt[6]{\frac{81}{4}}$

Step 1

Rewrite $81$ as $9^{2}$ .

$\sqrt[6]{\frac{9^{2}}{4}}$

Step 2

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

$\sqrt[6]{\frac{9^{2}}{2^{2}}}$

Step 3

Rewrite $\frac{9^{2}}{2^{2}}$ as ${(\frac{9}{2})}^{2}$ .

$\sqrt[6]{{(\frac{9}{2})}^{2}}$

Step 4

Rewrite $\sqrt[6]{{(\frac{9}{2})}^{2}}$ as $\sqrt[3]{\sqrt{{(\frac{9}{2})}^{2}}}$ .

$\sqrt[3]{\sqrt{{(\frac{9}{2})}^{2}}}$

Step 5

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

$\sqrt[3]{\frac{9}{2}}$

Step 6

Rewrite $\sqrt[3]{\frac{9}{2}}$ as $\frac{\sqrt[3]{9}}{\sqrt[3]{2}}$ .

$\frac{\sqrt[3]{9}}{\sqrt[3]{2}}$

Step 7

Multiply $\frac{\sqrt[3]{9}}{\sqrt[3]{2}}$ by $\frac{{\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{2}}$ .

$\frac{\sqrt[3]{9}}{\sqrt[3]{2}} \cdot \frac{{\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{2}}$

Step 8

Combine and simplify the denominator.

Multiply $\frac{\sqrt[3]{9}}{\sqrt[3]{2}}$ by $\frac{{\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{2}}$ .

$\frac{\sqrt[3]{9} {\sqrt[3]{2}}^{2}}{\sqrt[3]{2} {\sqrt[3]{2}}^{2}}$

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

$\frac{\sqrt[3]{9} {\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{1} {\sqrt[3]{2}}^{2}}$

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

$\frac{\sqrt[3]{9} {\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{1 + 2}}$

Add $1$ and $2$ .

$\frac{\sqrt[3]{9} {\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{3}}$

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

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

$\frac{\sqrt[3]{9} {\sqrt[3]{2}}^{2}}{{(2^{\frac{1}{3}})}^{3}}$

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

$\frac{\sqrt[3]{9} {\sqrt[3]{2}}^{2}}{2^{\frac{1}{3} \cdot 3}}$

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

$\frac{\sqrt[3]{9} {\sqrt[3]{2}}^{2}}{2^{\frac{3}{3}}}$

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{\sqrt[3]{9} {\sqrt[3]{2}}^{2}}{2^{\frac{3}{3}}}$

Rewrite the expression.

$\frac{\sqrt[3]{9} {\sqrt[3]{2}}^{2}}{2^{1}}$

Evaluate the exponent.

$\frac{\sqrt[3]{9} {\sqrt[3]{2}}^{2}}{2}$

Step 9

Simplify the numerator.

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

$\frac{\sqrt[3]{9} \sqrt[3]{2^{2}}}{2}$

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

$\frac{\sqrt[3]{9} \sqrt[3]{4}}{2}$

Step 10

Simplify the numerator.

Combine using the product rule for radicals.

$\frac{\sqrt[3]{9 \cdot 4}}{2}$

Multiply $9$ by $4$ .

$\frac{\sqrt[3]{36}}{2}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt[3]{36}}{2}$

Decimal Form:

$1.65096362 \dots$

Evaluate sixth root of 81/4

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