Algebra Examples

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

Step 1

Combine $\sqrt[3]{9}$ and $\sqrt[3]{6}$ into a single radical.

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

Step 2

Cancel the common factor of $9$ and $6$ .

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Step 2.1

Factor $3$ out of $9$ .

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

Step 2.2

Cancel the common factors.

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Step 2.2.1

Factor $3$ out of $6$ .

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

Step 2.2.2

Cancel the common factor.

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

Step 2.2.3

Rewrite the expression.

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

Step 3

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

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

Step 4

Multiply $\frac{\sqrt[3]{3}}{\sqrt[3]{2}}$ by $\frac{{\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{2}}$ .

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

Step 5

Combine and simplify the denominator.

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Step 5.1

Multiply $\frac{\sqrt[3]{3}}{\sqrt[3]{2}}$ by $\frac{{\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{2}}$ .

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

Step 5.2

Raise $\sqrt[3]{2}$ to the power of $1$ .

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

Step 5.3

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

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

Step 5.4

Add $1$ and $2$ .

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

Step 5.5

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

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Step 5.5.1

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

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

Step 5.5.2

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

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

Step 5.5.3

Combine $\frac{1}{3}$ and $3$ .

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

Step 5.5.4

Cancel the common factor of $3$ .

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Step 5.5.4.1

Cancel the common factor.

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

Step 5.5.4.2

Rewrite the expression.

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

Step 5.5.5

Evaluate the exponent.

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

Step 6

Simplify the numerator.

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Step 6.1

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

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

Step 6.2

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

$\frac{\sqrt[3]{3} \sqrt[3]{4}}{2}$

Step 7

Simplify the numerator.

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Step 7.1

Combine using the product rule for radicals.

$\frac{\sqrt[3]{3 \cdot 4}}{2}$

Step 7.2

Multiply $3$ by $4$ .

$\frac{\sqrt[3]{12}}{2}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt[3]{12}}{2}$

Decimal Form:

$1.14471424 \dots$