Simplify square root of 3* cube root of 18

$\sqrt{3} \cdot \sqrt[3]{18}$

Step 1

Multiply $\sqrt{3} \sqrt[3]{18}$ .

Rewrite the expression using the least common index of $6$ .

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

$3^{\frac{1}{2}} \sqrt[3]{18}$

Rewrite $3^{\frac{1}{2}}$ as $3^{\frac{3}{6}}$ .

$3^{\frac{3}{6}} \sqrt[3]{18}$

Rewrite $3^{\frac{3}{6}}$ as $\sqrt[6]{3^{3}}$ .

$\sqrt[6]{3^{3}} \sqrt[3]{18}$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{18}$ as $18^{\frac{1}{3}}$ .

$\sqrt[6]{3^{3}} \cdot 18^{\frac{1}{3}}$

Rewrite $18^{\frac{1}{3}}$ as $18^{\frac{2}{6}}$ .

$\sqrt[6]{3^{3}} \cdot 18^{\frac{2}{6}}$

Rewrite $18^{\frac{2}{6}}$ as $\sqrt[6]{18^{2}}$ .

$\sqrt[6]{3^{3}} \sqrt[6]{18^{2}}$

Combine using the product rule for radicals.

$\sqrt[6]{3^{3} \cdot 18^{2}}$

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

$\sqrt[6]{27 \cdot 18^{2}}$

Raise $18$ to the power of $2$ .

$\sqrt[6]{27 \cdot 324}$

Multiply $27$ by $324$ .

$\sqrt[6]{8748}$

Step 2

Rewrite $8748$ as $3^{6} \cdot 12$ .

Factor $729$ out of $8748$ .

$\sqrt[6]{729 (12)}$

Rewrite $729$ as $3^{6}$ .

$\sqrt[6]{3^{6} \cdot 12}$

Step 3

Pull terms out from under the radical.

$3 \sqrt[6]{12}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$3 \sqrt[6]{12}$

Decimal Form:

$4.53925724 \dots$