Evaluate square root of 3( fourth root of 3)

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

Step 1

Multiply $\sqrt{3} (\sqrt[4]{3})$ .

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

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

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

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

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

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

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

Combine using the product rule for radicals.

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

Multiply $3^{2}$ by $3$ by adding the exponents.

Multiply $3^{2}$ by $3$ .

Raise $3$ to the power of $1$ .

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

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

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

Add $2$ and $1$ .

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

Step 2

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

$\sqrt[4]{27}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\sqrt[4]{27}$

Decimal Form:

$2.27950705 \dots$