Simplify the Radical Expression ( square root of 3)/( fourth root of 3)

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

Step 1

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

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

Step 2

Combine and simplify the denominator.

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

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

Raise $\sqrt[4]{3}$ to the power of $1$ .

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

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

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

Add $1$ and $3$ .

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

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

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{3}$ as $3^{\frac{1}{4}}$ .

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

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

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

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

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

Cancel the common factor of $4$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Step 3

Simplify the numerator.

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

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

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

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

Step 4

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 $3^{\frac{1}{2}}$ .

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

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

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

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

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

Combine using the product rule for radicals.

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

Rewrite $27$ as $3^{3}$ .

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

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

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

Add $2$ and $3$ .

$\frac{\sqrt[4]{3^{5}}}{3}$

Step 5

Simplify the numerator.

Raise $3$ to the power of $5$ .

$\frac{\sqrt[4]{243}}{3}$

Rewrite $243$ as $3^{4} \cdot 3$ .

Factor $81$ out of $243$ .

$\frac{\sqrt[4]{81 (3)}}{3}$

Rewrite $81$ as $3^{4}$ .

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

Pull terms out from under the radical.

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

The absolute value is the distance between a number and zero. The distance between $0$ and $3$ is $3$ .

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

Step 6

Cancel the common factor of $3$ .

Cancel the common factor.

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

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

$\sqrt[4]{3}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\sqrt[4]{3}$

Decimal Form:

$1.31607401 \dots$

Simplify the Radical Expression ( square root of 3)/( fourth root of 3)

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