Precalculus Examples

$\frac{6 \sqrt[3]{4 v^{7}}}{\sqrt[3]{14 v}}$

Step 1

Combine $\sqrt[3]{4 v^{7}}$ and $\sqrt[3]{14 v}$ into a single radical.

$6 \sqrt[3]{\frac{4 v^{7}}{14 v}}$

Step 2

Reduce the expression $\frac{4 v^{7}}{14 v}$ by cancelling the common factors.

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Step 2.1

Factor $2$ out of $4 v^{7}$ .

$6 \sqrt[3]{\frac{2 (2 v^{7})}{14 v}}$

Step 2.2

Factor $2$ out of $14 v$ .

$6 \sqrt[3]{\frac{2 (2 v^{7})}{2 (7 v)}}$

Step 2.3

Cancel the common factor.

$6 \sqrt[3]{\frac{2 (2 v^{7})}{2 (7 v)}}$

Step 2.4

Rewrite the expression.

$6 \sqrt[3]{\frac{2 v^{7}}{7 v}}$

Step 3

Cancel the common factor of $v^{7}$ and $v$ .

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Step 3.1

Factor $v$ out of $2 v^{7}$ .

$6 \sqrt[3]{\frac{v (2 v^{6})}{7 v}}$

Step 3.2

Cancel the common factors.

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Step 3.2.1

Factor $v$ out of $7 v$ .

$6 \sqrt[3]{\frac{v (2 v^{6})}{v \cdot 7}}$

Step 3.2.2

Cancel the common factor.

$6 \sqrt[3]{\frac{v (2 v^{6})}{v \cdot 7}}$

Step 3.2.3

Rewrite the expression.

$6 \sqrt[3]{\frac{2 v^{6}}{7}}$

Step 4

Rewrite $\sqrt[3]{\frac{2 v^{6}}{7}}$ as $\frac{\sqrt[3]{2 v^{6}}}{\sqrt[3]{7}}$ .

$6 \frac{\sqrt[3]{2 v^{6}}}{\sqrt[3]{7}}$

Step 5

Simplify the numerator.

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Step 5.1

Rewrite $2 v^{6}$ as ${(v^{2})}^{3} \cdot 2$ .

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Step 5.1.1

Rewrite $v^{6}$ as ${(v^{2})}^{3}$ .

$6 \frac{\sqrt[3]{2 {(v^{2})}^{3}}}{\sqrt[3]{7}}$

Step 5.1.2

Reorder $2$ and ${(v^{2})}^{3}$ .

$6 \frac{\sqrt[3]{{(v^{2})}^{3} \cdot 2}}{\sqrt[3]{7}}$

Step 5.2

Pull terms out from under the radical.

$6 \frac{v^{2} \sqrt[3]{2}}{\sqrt[3]{7}}$

Step 6

Multiply $\frac{v^{2} \sqrt[3]{2}}{\sqrt[3]{7}}$ by $\frac{{\sqrt[3]{7}}^{2}}{{\sqrt[3]{7}}^{2}}$ .

$6 (\frac{v^{2} \sqrt[3]{2}}{\sqrt[3]{7}} \cdot \frac{{\sqrt[3]{7}}^{2}}{{\sqrt[3]{7}}^{2}})$

Step 7

Combine and simplify the denominator.

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Step 7.1

Multiply $\frac{v^{2} \sqrt[3]{2}}{\sqrt[3]{7}}$ by $\frac{{\sqrt[3]{7}}^{2}}{{\sqrt[3]{7}}^{2}}$ .

$6 \frac{v^{2} \sqrt[3]{2} {\sqrt[3]{7}}^{2}}{\sqrt[3]{7} {\sqrt[3]{7}}^{2}}$

Step 7.2

Raise $\sqrt[3]{7}$ to the power of $1$ .

$6 \frac{v^{2} \sqrt[3]{2} {\sqrt[3]{7}}^{2}}{{\sqrt[3]{7}}^{1} {\sqrt[3]{7}}^{2}}$

Step 7.3

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

$6 \frac{v^{2} \sqrt[3]{2} {\sqrt[3]{7}}^{2}}{{\sqrt[3]{7}}^{1 + 2}}$

Step 7.4

Add $1$ and $2$ .

$6 \frac{v^{2} \sqrt[3]{2} {\sqrt[3]{7}}^{2}}{{\sqrt[3]{7}}^{3}}$

Step 7.5

Rewrite ${\sqrt[3]{7}}^{3}$ as $7$ .

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Step 7.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{7}$ as $7^{\frac{1}{3}}$ .

$6 \frac{v^{2} \sqrt[3]{2} {\sqrt[3]{7}}^{2}}{{(7^{\frac{1}{3}})}^{3}}$

Step 7.5.2

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

$6 \frac{v^{2} \sqrt[3]{2} {\sqrt[3]{7}}^{2}}{7^{\frac{1}{3} \cdot 3}}$

Step 7.5.3

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

$6 \frac{v^{2} \sqrt[3]{2} {\sqrt[3]{7}}^{2}}{7^{\frac{3}{3}}}$

Step 7.5.4

Cancel the common factor of $3$ .

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Step 7.5.4.1

Cancel the common factor.

$6 \frac{v^{2} \sqrt[3]{2} {\sqrt[3]{7}}^{2}}{7^{\frac{3}{3}}}$

Step 7.5.4.2

Rewrite the expression.

$6 \frac{v^{2} \sqrt[3]{2} {\sqrt[3]{7}}^{2}}{7^{1}}$

Step 7.5.5

Evaluate the exponent.

$6 \frac{v^{2} \sqrt[3]{2} {\sqrt[3]{7}}^{2}}{7}$

Step 8

Simplify the numerator.

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Step 8.1

Rewrite ${\sqrt[3]{7}}^{2}$ as $\sqrt[3]{7^{2}}$ .

$6 \frac{v^{2} \sqrt[3]{2} \sqrt[3]{7^{2}}}{7}$

Step 8.2

Raise $7$ to the power of $2$ .

$6 \frac{v^{2} \sqrt[3]{2} \sqrt[3]{49}}{7}$

Step 8.3

Combine exponents.

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Step 8.3.1

Combine using the product rule for radicals.

$6 \frac{v^{2} \sqrt[3]{49 \cdot 2}}{7}$

Step 8.3.2

Multiply $49$ by $2$ .

$6 \frac{v^{2} \sqrt[3]{98}}{7}$

Step 9

Combine $6$ and $\frac{v^{2} \sqrt[3]{98}}{7}$ .

$\frac{6 v^{2} \sqrt[3]{98}}{7}$