Simplify square root of 3* cube root of 9

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

Step 1

Multiply $\sqrt{3} \sqrt[3]{9}$ .

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

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

$3^{\frac{1}{2}} \sqrt[3]{9}$

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

$3^{\frac{3}{6}} \sqrt[3]{9}$

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

$\sqrt[6]{3^{3}} \sqrt[3]{9}$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{9}$ as $9^{\frac{1}{3}}$ .

$\sqrt[6]{3^{3}} \cdot 9^{\frac{1}{3}}$

Rewrite $9^{\frac{1}{3}}$ as $9^{\frac{2}{6}}$ .

$\sqrt[6]{3^{3}} \cdot 9^{\frac{2}{6}}$

Rewrite $9^{\frac{2}{6}}$ as $\sqrt[6]{9^{2}}$ .

$\sqrt[6]{3^{3}} \sqrt[6]{9^{2}}$

Combine using the product rule for radicals.

$\sqrt[6]{3^{3} \cdot 9^{2}}$

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

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

Multiply the exponents in ${(3^{2})}^{2}$ .

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

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

Multiply $2$ by $2$ .

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

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

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

Add $3$ and $4$ .

$\sqrt[6]{3^{7}}$

Step 2

Raise $3$ to the power of $7$ .

$\sqrt[6]{2187}$

Step 3

Rewrite $2187$ as $3^{6} \cdot 3$ .

Factor $729$ out of $2187$ .

$\sqrt[6]{729 (3)}$

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

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

Step 4

Pull terms out from under the radical.

$3 \sqrt[6]{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$3 \sqrt[6]{3}$

Decimal Form:

$3.60281086 \dots$

Simplify square root of 3* cube root of 9

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