Simplify square root of 3/( fourth root of 3)

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

Step 1

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

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

Step 2

Combine and simplify the denominator.

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

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

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

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

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

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

Add $1$ and $3$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $4$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Step 3

Cancel the common factor of $3$ .

Cancel the common factor.

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

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

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

Step 4

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

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

Step 5

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

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

Step 6

Rewrite $\sqrt{\sqrt[4]{27}}$ as $\sqrt[8]{27}$ .

$\sqrt[8]{27}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\sqrt[8]{27}$

Decimal Form:

$1.50980364 \dots$

Simplify square root of 3/( fourth root of 3)

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