Evaluate 324^(1/4)* fourth root of 1/4

$324^{\frac{1}{4}} \cdot \sqrt[4]{\frac{1}{4}}$

Rewrite $1$ as $1^{2}$ .

$324^{\frac{1}{4}} \cdot \sqrt[4]{\frac{1^{2}}{4}}$

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

$324^{\frac{1}{4}} \cdot \sqrt[4]{\frac{1^{2}}{2^{2}}}$

Rewrite $\frac{1^{2}}{2^{2}}$ as ${(\frac{1}{2})}^{2}$ .

$324^{\frac{1}{4}} \cdot \sqrt[4]{{(\frac{1}{2})}^{2}}$

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

$324^{\frac{1}{4}} \cdot \sqrt{\sqrt{{(\frac{1}{2})}^{2}}}$

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

$324^{\frac{1}{4}} \cdot \sqrt{\frac{1}{2}}$

Rewrite $\sqrt{\frac{1}{2}}$ as $\frac{\sqrt{1}}{\sqrt{2}}$ .

$324^{\frac{1}{4}} \cdot \frac{\sqrt{1}}{\sqrt{2}}$

Any root of $1$ is $1$ .

$324^{\frac{1}{4}} \cdot \frac{1}{\sqrt{2}}$

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$324^{\frac{1}{4}} \cdot (\frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}})$

Combine and simplify the denominator.

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$324^{\frac{1}{4}} \cdot \frac{\sqrt{2}}{\sqrt{2} \sqrt{2}}$

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

$324^{\frac{1}{4}} \cdot \frac{\sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

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

$324^{\frac{1}{4}} \cdot \frac{\sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

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

$324^{\frac{1}{4}} \cdot \frac{\sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Add $1$ and $1$ .

$324^{\frac{1}{4}} \cdot \frac{\sqrt{2}}{{\sqrt{2}}^{2}}$

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

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

$324^{\frac{1}{4}} \cdot \frac{\sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

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

$324^{\frac{1}{4}} \cdot \frac{\sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

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

$324^{\frac{1}{4}} \cdot \frac{\sqrt{2}}{2^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$324^{\frac{1}{4}} \cdot \frac{\sqrt{2}}{2^{\frac{2}{2}}}$

Rewrite the expression.

$324^{\frac{1}{4}} \cdot \frac{\sqrt{2}}{2^{1}}$

Evaluate the exponent.

$324^{\frac{1}{4}} \cdot \frac{\sqrt{2}}{2}$

Combine $324^{\frac{1}{4}}$ and $\frac{\sqrt{2}}{2}$ .

$\frac{324^{\frac{1}{4}} \sqrt{2}}{2}$

The result can be shown in multiple forms.

Exact Form:

$\frac{324^{\frac{1}{4}} \sqrt{2}}{2}$

Decimal Form:

$3$

Evaluate 324^(1/4)* fourth root of 1/4

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