Algebra Examples

$\sqrt[3]{\frac{p^{5} q^{3}}{40}}$

Step 1

Rewrite $\frac{p^{5} q^{3}}{40}$ as ${(\frac{p q}{2})}^{3} \frac{p^{2}}{5}$ .

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Step 1.1

Factor the perfect power ${(p q)}^{3}$ out of $p^{5} q^{3}$ .

$\sqrt[3]{\frac{{(p q)}^{3} p^{2}}{40}}$

Step 1.2

Factor the perfect power $2^{3}$ out of $40$ .

$\sqrt[3]{\frac{{(p q)}^{3} p^{2}}{2^{3} \cdot 5}}$

Step 1.3

Rearrange the fraction $\frac{{(p q)}^{3} p^{2}}{2^{3} \cdot 5}$ .

$\sqrt[3]{{(\frac{p q}{2})}^{3} \frac{p^{2}}{5}}$

Step 2

Pull terms out from under the radical.

$\frac{p q}{2} \sqrt[3]{\frac{p^{2}}{5}}$

Step 3

Rewrite $\sqrt[3]{\frac{p^{2}}{5}}$ as $\frac{\sqrt[3]{p^{2}}}{\sqrt[3]{5}}$ .

$\frac{p q}{2} \cdot \frac{\sqrt[3]{p^{2}}}{\sqrt[3]{5}}$

Step 4

Combine.

$\frac{p q \sqrt[3]{p^{2}}}{2 \sqrt[3]{5}}$

Step 5

Multiply $\frac{p q \sqrt[3]{p^{2}}}{2 \sqrt[3]{5}}$ by $\frac{{\sqrt[3]{5}}^{2}}{{\sqrt[3]{5}}^{2}}$ .

$\frac{p q \sqrt[3]{p^{2}}}{2 \sqrt[3]{5}} \cdot \frac{{\sqrt[3]{5}}^{2}}{{\sqrt[3]{5}}^{2}}$

Step 6

Combine and simplify the denominator.

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Step 6.1

Multiply $\frac{p q \sqrt[3]{p^{2}}}{2 \sqrt[3]{5}}$ by $\frac{{\sqrt[3]{5}}^{2}}{{\sqrt[3]{5}}^{2}}$ .

$\frac{p q \sqrt[3]{p^{2}} {\sqrt[3]{5}}^{2}}{2 \sqrt[3]{5} {\sqrt[3]{5}}^{2}}$

Step 6.2

Move $\sqrt[3]{5}$ .

$\frac{p q \sqrt[3]{p^{2}} {\sqrt[3]{5}}^{2}}{2 (\sqrt[3]{5} {\sqrt[3]{5}}^{2})}$

Step 6.3

Raise $\sqrt[3]{5}$ to the power of $1$ .

$\frac{p q \sqrt[3]{p^{2}} {\sqrt[3]{5}}^{2}}{2 ({\sqrt[3]{5}}^{1} {\sqrt[3]{5}}^{2})}$

Step 6.4

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

$\frac{p q \sqrt[3]{p^{2}} {\sqrt[3]{5}}^{2}}{2 {\sqrt[3]{5}}^{1 + 2}}$

Step 6.5

Add $1$ and $2$ .

$\frac{p q \sqrt[3]{p^{2}} {\sqrt[3]{5}}^{2}}{2 {\sqrt[3]{5}}^{3}}$

Step 6.6

Rewrite ${\sqrt[3]{5}}^{3}$ as $5$ .

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Step 6.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{5}$ as $5^{\frac{1}{3}}$ .

$\frac{p q \sqrt[3]{p^{2}} {\sqrt[3]{5}}^{2}}{2 {(5^{\frac{1}{3}})}^{3}}$

Step 6.6.2

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

$\frac{p q \sqrt[3]{p^{2}} {\sqrt[3]{5}}^{2}}{2 \cdot 5^{\frac{1}{3} \cdot 3}}$

Step 6.6.3

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

$\frac{p q \sqrt[3]{p^{2}} {\sqrt[3]{5}}^{2}}{2 \cdot 5^{\frac{3}{3}}}$

Step 6.6.4

Cancel the common factor of $3$ .

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Step 6.6.4.1

Cancel the common factor.

$\frac{p q \sqrt[3]{p^{2}} {\sqrt[3]{5}}^{2}}{2 \cdot 5^{\frac{3}{3}}}$

Step 6.6.4.2

Rewrite the expression.

$\frac{p q \sqrt[3]{p^{2}} {\sqrt[3]{5}}^{2}}{2 \cdot 5^{1}}$

Step 6.6.5

Evaluate the exponent.

$\frac{p q \sqrt[3]{p^{2}} {\sqrt[3]{5}}^{2}}{2 \cdot 5}$

Step 7

Simplify the numerator.

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Step 7.1

Rewrite ${\sqrt[3]{5}}^{2}$ as $\sqrt[3]{5^{2}}$ .

$\frac{p q \sqrt[3]{p^{2}} \sqrt[3]{5^{2}}}{2 \cdot 5}$

Step 7.2

Raise $5$ to the power of $2$ .

$\frac{p q \sqrt[3]{p^{2}} \sqrt[3]{25}}{2 \cdot 5}$

Step 7.3

Combine using the product rule for radicals.

$\frac{p q \sqrt[3]{25 p^{2}}}{2 \cdot 5}$

Step 8

Multiply $2$ by $5$ .

$\frac{p q \sqrt[3]{25 p^{2}}}{10}$