Simplify cube root of 2* square root of 3

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

Step 1

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

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

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

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

$2^{\frac{2}{6}} \sqrt{3}$

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

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

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

$\sqrt[6]{2^{2}} \cdot 3^{\frac{1}{2}}$

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

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

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

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

Step 2

Combine using the product rule for radicals.

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

Step 3

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

$\sqrt[6]{4 \cdot 3^{3}}$

Step 4

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

$\sqrt[6]{4 \cdot 27}$

Step 5

Multiply $4$ by $27$ .

$\sqrt[6]{108}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\sqrt[6]{108}$

Decimal Form:

$2.18224727 \dots$