Simplify fourth root of 2/9

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

Step 1

Rewrite $\sqrt[4]{\frac{2}{9}}$ as $\frac{\sqrt[4]{2}}{\sqrt[4]{9}}$ .

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

Step 2

Simplify the denominator.

Rewrite $9$ as $3^{2}$ .

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

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

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

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

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

Step 3

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

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

Step 4

Combine and simplify the denominator.

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

Combine $\frac{1}{2}$ and $2$ .

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Step 5

Simplify the numerator.

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

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

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

Rewrite $3^{\frac{1}{2}}$ as $3^{\frac{2}{4}}$ .

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

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

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

Combine using the product rule for radicals.

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

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

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

Step 6

Multiply $2$ by $9$ .

$\frac{\sqrt[4]{18}}{3}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt[4]{18}}{3}$

Decimal Form:

$0.68658904 \dots$

Simplify fourth root of 2/9

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