Evaluate ( fourth root of 4)/( fourth root of 27)

$\frac{\sqrt[4]{4}}{\sqrt[4]{27}}$

Step 1

Simplify the numerator.

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

$\frac{\sqrt[4]{2^{2}}}{\sqrt[4]{27}}$

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

$\frac{\sqrt{\sqrt{2^{2}}}}{\sqrt[4]{27}}$

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

$\frac{\sqrt{2}}{\sqrt[4]{27}}$

Step 2

Multiply $\frac{\sqrt{2}}{\sqrt[4]{27}}$ by $\frac{{\sqrt[4]{27}}^{3}}{{\sqrt[4]{27}}^{3}}$ .

$\frac{\sqrt{2}}{\sqrt[4]{27}} \cdot \frac{{\sqrt[4]{27}}^{3}}{{\sqrt[4]{27}}^{3}}$

Step 3

Combine and simplify the denominator.

Multiply $\frac{\sqrt{2}}{\sqrt[4]{27}}$ by $\frac{{\sqrt[4]{27}}^{3}}{{\sqrt[4]{27}}^{3}}$ .

$\frac{\sqrt{2} {\sqrt[4]{27}}^{3}}{\sqrt[4]{27} {\sqrt[4]{27}}^{3}}$

Raise $\sqrt[4]{27}$ to the power of $1$ .

$\frac{\sqrt{2} {\sqrt[4]{27}}^{3}}{{\sqrt[4]{27}}^{1} {\sqrt[4]{27}}^{3}}$

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

$\frac{\sqrt{2} {\sqrt[4]{27}}^{3}}{{\sqrt[4]{27}}^{1 + 3}}$

Add $1$ and $3$ .

$\frac{\sqrt{2} {\sqrt[4]{27}}^{3}}{{\sqrt[4]{27}}^{4}}$

Rewrite ${\sqrt[4]{27}}^{4}$ as $27$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{27}$ as $27^{\frac{1}{4}}$ .

$\frac{\sqrt{2} {\sqrt[4]{27}}^{3}}{{(27^{\frac{1}{4}})}^{4}}$

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

$\frac{\sqrt{2} {\sqrt[4]{27}}^{3}}{27^{\frac{1}{4} \cdot 4}}$

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

$\frac{\sqrt{2} {\sqrt[4]{27}}^{3}}{27^{\frac{4}{4}}}$

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{\sqrt{2} {\sqrt[4]{27}}^{3}}{27^{\frac{4}{4}}}$

Rewrite the expression.

$\frac{\sqrt{2} {\sqrt[4]{27}}^{3}}{27^{1}}$

Evaluate the exponent.

$\frac{\sqrt{2} {\sqrt[4]{27}}^{3}}{27}$

Step 4

Simplify the numerator.

Rewrite ${\sqrt[4]{27}}^{3}$ as ${(27^{3})}^{\frac{1}{4}}$ .

$\frac{\sqrt{2} \sqrt[4]{27^{3}}}{27}$

Raise $27$ to the power of $3$ .

$\frac{\sqrt{2} \sqrt[4]{19683}}{27}$

Rewrite $19683$ as $9^{4} \cdot 3$ .

Factor $6561$ out of $19683$ .

$\frac{\sqrt{2} \sqrt[4]{6561 (3)}}{27}$

Rewrite $6561$ as $9^{4}$ .

$\frac{\sqrt{2} \sqrt[4]{9^{4} \cdot 3}}{27}$

Pull terms out from under the radical.

$\frac{\sqrt{2} \cdot 9 \sqrt[4]{3}}{27}$

Combine exponents.

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

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

$\frac{9 (2^{\frac{1}{2}} \sqrt[4]{3})}{27}$

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

$\frac{9 (2^{\frac{2}{4}} \sqrt[4]{3})}{27}$

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

$\frac{9 (\sqrt[4]{2^{2}} \sqrt[4]{3})}{27}$

Combine using the product rule for radicals.

$\frac{9 \sqrt[4]{2^{2} \cdot 3}}{27}$

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

$\frac{9 \sqrt[4]{4 \cdot 3}}{27}$

Multiply $4$ by $3$ .

$\frac{9 \sqrt[4]{12}}{27}$

Step 5

Cancel the common factor of $9$ and $27$ .

Factor $9$ out of $9 \sqrt[4]{12}$ .

$\frac{9 (\sqrt[4]{12})}{27}$

Cancel the common factors.

Factor $9$ out of $27$ .

$\frac{9 \sqrt[4]{12}}{9 \cdot 3}$

Cancel the common factor.

$\frac{9 \sqrt[4]{12}}{9 \cdot 3}$

Rewrite the expression.

$\frac{\sqrt[4]{12}}{3}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt[4]{12}}{3}$

Decimal Form:

$0.62040323 \dots$

Evaluate ( fourth root of 4)/( fourth root of 27)

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