Simplify ( cube root of 9)÷( square root of 3)

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

Step 1

Rewrite the division as a fraction.

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

Step 2

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

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

Step 3

Combine and simplify the denominator.

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

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

Raise $\sqrt{3}$ to the power of $1$ .

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

Raise $\sqrt{3}$ to the power of $1$ .

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

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

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

Add $1$ and $1$ .

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

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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

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

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

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

Combine $\frac{1}{2}$ and $2$ .

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Step 4

Simplify the numerator.

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

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

$\frac{9^{\frac{1}{3}} \sqrt{3}}{3}$

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

$\frac{9^{\frac{2}{6}} \sqrt{3}}{3}$

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

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

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

$\frac{\sqrt[6]{9^{2}} \cdot 3^{\frac{1}{2}}}{3}$

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

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

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

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

Combine using the product rule for radicals.

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

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

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

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

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

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

Multiply $2$ by $2$ .

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

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

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

Add $4$ and $3$ .

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

Step 5

Simplify the numerator.

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

$\frac{\sqrt[6]{2187}}{3}$

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

Factor $729$ out of $2187$ .

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

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

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

Pull terms out from under the radical.

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

Step 6

Cancel the common factor of $3$ .

Cancel the common factor.

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

Divide $\sqrt[6]{3}$ by $1$ .

$\sqrt[6]{3}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\sqrt[6]{3}$

Decimal Form:

$1.20093695 \dots$

Simplify ( cube root of 9)÷( square root of 3)

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