Simplify cube root of 6* fourth root of 9

$\sqrt[3]{6} \cdot \sqrt[4]{9}$

Step 1

Rewrite $9$ as $3^{2}$ .

$\sqrt[3]{6} \cdot \sqrt[4]{3^{2}}$

Step 2

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

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

Step 3

Pull terms out from under the radical, assuming positive real numbers.

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

Step 4

Multiply $\sqrt[3]{6} \sqrt{3}$ .

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

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{6}$ as $6^{\frac{1}{3}}$ .

$6^{\frac{1}{3}} \sqrt{3}$

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

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

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

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

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

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

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

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

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

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

Combine using the product rule for radicals.

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

Raise $6$ to the power of $2$ .

$\sqrt[6]{36 \cdot 3^{3}}$

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

$\sqrt[6]{36 \cdot 27}$

Multiply $36$ by $27$ .

$\sqrt[6]{972}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\sqrt[6]{972}$

Decimal Form:

$3.14734519 \dots$