Algebra Examples

$\sqrt[3]{32} \div \sqrt[3]{18}$

Step 1

Rewrite the division as a fraction.

$\frac{\sqrt[3]{32}}{\sqrt[3]{18}}$

Step 2

Combine $\sqrt[3]{32}$ and $\sqrt[3]{18}$ into a single radical.

$\sqrt[3]{\frac{32}{18}}$

Step 3

Cancel the common factor of $32$ and $18$ .

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Step 3.1

Factor $2$ out of $32$ .

$\sqrt[3]{\frac{2 (16)}{18}}$

Step 3.2

Cancel the common factors.

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Step 3.2.1

Factor $2$ out of $18$ .

$\sqrt[3]{\frac{2 \cdot 16}{2 \cdot 9}}$

Step 3.2.2

Cancel the common factor.

$\sqrt[3]{\frac{2 \cdot 16}{2 \cdot 9}}$

Step 3.2.3

Rewrite the expression.

$\sqrt[3]{\frac{16}{9}}$

Step 4

Rewrite $\sqrt[3]{\frac{16}{9}}$ as $\frac{\sqrt[3]{16}}{\sqrt[3]{9}}$ .

$\frac{\sqrt[3]{16}}{\sqrt[3]{9}}$

Step 5

Simplify the numerator.

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Step 5.1

Rewrite $16$ as $2^{3} \cdot 2$ .

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Step 5.1.1

Factor $8$ out of $16$ .

$\frac{\sqrt[3]{8 (2)}}{\sqrt[3]{9}}$

Step 5.1.2

Rewrite $8$ as $2^{3}$ .

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

Step 5.2

Pull terms out from under the radical.

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

Step 6

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

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

Step 7

Combine and simplify the denominator.

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Step 7.1

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

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

Step 7.2

Raise $\sqrt[3]{9}$ to the power of $1$ .

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

Step 7.3

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

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

Step 7.4

Add $1$ and $2$ .

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

Step 7.5

Rewrite ${\sqrt[3]{9}}^{3}$ as $9$ .

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Step 7.5.1

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

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

Step 7.5.2

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

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

Step 7.5.3

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

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

Step 7.5.4

Cancel the common factor of $3$ .

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Step 7.5.4.1

Cancel the common factor.

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

Step 7.5.4.2

Rewrite the expression.

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

Step 7.5.5

Evaluate the exponent.

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

Step 8

Simplify the numerator.

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Step 8.1

Rewrite ${\sqrt[3]{9}}^{2}$ as $\sqrt[3]{9^{2}}$ .

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

Step 8.2

Raise $9$ to the power of $2$ .

$\frac{2 \sqrt[3]{2} \sqrt[3]{81}}{9}$

Step 8.3

Rewrite $81$ as $3^{3} \cdot 3$ .

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Step 8.3.1

Factor $27$ out of $81$ .

$\frac{2 \sqrt[3]{2} \sqrt[3]{27 (3)}}{9}$

Step 8.3.2

Rewrite $27$ as $3^{3}$ .

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

Step 8.4

Pull terms out from under the radical.

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

Step 8.5

Combine exponents.

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Step 8.5.1

Multiply $3$ by $2$ .

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

Step 8.5.2

Combine using the product rule for radicals.

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

Step 8.5.3

Multiply $3$ by $2$ .

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

Step 9

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

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Step 9.1

Factor $3$ out of $6 \sqrt[3]{6}$ .

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

Step 9.2

Cancel the common factors.

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Step 9.2.1

Factor $3$ out of $9$ .

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

Step 9.2.2

Cancel the common factor.

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

Step 9.2.3

Rewrite the expression.

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

Step 10

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$1.21141372 \dots$