Algebra Examples

$\frac{\sqrt[3]{3} \cdot \sqrt[3]{18}}{\sqrt[6]{2} \cdot \sqrt[6]{2}}$

Step 1

Simplify the numerator.

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Step 1.1

Combine using the product rule for radicals.

$\frac{\sqrt[3]{3 \cdot 18}}{\sqrt[6]{2} \cdot \sqrt[6]{2}}$

Step 1.2

Multiply $3$ by $18$ .

$\frac{\sqrt[3]{54}}{\sqrt[6]{2} \cdot \sqrt[6]{2}}$

Step 2

Simplify the denominator.

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Step 2.1

Raise $\sqrt[6]{2}$ to the power of $1$ .

$\frac{\sqrt[3]{54}}{{\sqrt[6]{2}}^{1} \sqrt[6]{2}}$

Step 2.2

Raise $\sqrt[6]{2}$ to the power of $1$ .

$\frac{\sqrt[3]{54}}{{\sqrt[6]{2}}^{1} {\sqrt[6]{2}}^{1}}$

Step 2.3

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

$\frac{\sqrt[3]{54}}{{\sqrt[6]{2}}^{1 + 1}}$

Step 2.4

Add $1$ and $1$ .

$\frac{\sqrt[3]{54}}{{\sqrt[6]{2}}^{2}}$

Step 3

Simplify the numerator.

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Step 3.1

Rewrite $54$ as $3^{3} \cdot 2$ .

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Step 3.1.1

Factor $27$ out of $54$ .

$\frac{\sqrt[3]{27 (2)}}{{\sqrt[6]{2}}^{2}}$

Step 3.1.2

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

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

Step 3.2

Pull terms out from under the radical.

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

Step 4

Rewrite ${\sqrt[6]{2}}^{2}$ as $\sqrt[3]{2}$ .

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Step 4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[6]{2}$ as $2^{\frac{1}{6}}$ .

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

Step 4.2

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

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

Step 4.3

Combine $\frac{1}{6}$ and $2$ .

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

Step 4.4

Cancel the common factor of $2$ and $6$ .

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Step 4.4.1

Factor $2$ out of $2$ .

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

Step 4.4.2

Cancel the common factors.

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Step 4.4.2.1

Factor $2$ out of $6$ .

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

Step 4.4.2.2

Cancel the common factor.

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

Step 4.4.2.3

Rewrite the expression.

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

Step 4.5

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

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

Step 5

Cancel the common factor of $\sqrt[3]{2}$ .

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Step 5.1

Cancel the common factor.

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

Step 5.2

Divide $3$ by $1$ .

$3$