Algebra Examples

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

Step 1

Simplify the expression.

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Step 1.1

Apply the product rule to $\frac{3}{2 p^{2}}$ .

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

Step 1.2

Raise $3$ to the power of $2$ .

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

Step 2

Simplify the denominator.

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Step 2.1

Apply the product rule to $2 p^{2}$ .

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

Step 2.2

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

$\sqrt[3]{\frac{9}{4 {(p^{2})}^{2}}}$

Step 2.3

Multiply the exponents in ${(p^{2})}^{2}$ .

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Step 2.3.1

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

$\sqrt[3]{\frac{9}{4 p^{2 \cdot 2}}}$

Step 2.3.2

Multiply $2$ by $2$ .

$\sqrt[3]{\frac{9}{4 p^{4}}}$

Step 3

Rewrite $\frac{9}{4 p^{4}}$ as ${(\frac{1}{p})}^{3} \frac{9}{4 p}$ .

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Step 3.1

Factor the perfect power $1^{3}$ out of $9$ .

$\sqrt[3]{\frac{1^{3} \cdot 9}{4 p^{4}}}$

Step 3.2

Factor the perfect power $p^{3}$ out of $4 p^{4}$ .

$\sqrt[3]{\frac{1^{3} \cdot 9}{p^{3} \cdot (4 p)}}$

Step 3.3

Rearrange the fraction $\frac{1^{3} \cdot 9}{p^{3} \cdot (4 p)}$ .

$\sqrt[3]{{(\frac{1}{p})}^{3} \frac{9}{4 p}}$

Step 4

Pull terms out from under the radical.

$\frac{1}{p} \sqrt[3]{\frac{9}{4 p}}$

Step 5

Rewrite $\sqrt[3]{\frac{9}{4 p}}$ as $\frac{\sqrt[3]{9}}{\sqrt[3]{4 p}}$ .

$\frac{1}{p} \cdot \frac{\sqrt[3]{9}}{\sqrt[3]{4 p}}$

Step 6

Multiply $\frac{\sqrt[3]{9}}{\sqrt[3]{4 p}}$ by $\frac{{\sqrt[3]{4 p}}^{2}}{{\sqrt[3]{4 p}}^{2}}$ .

$\frac{1}{p} (\frac{\sqrt[3]{9}}{\sqrt[3]{4 p}} \cdot \frac{{\sqrt[3]{4 p}}^{2}}{{\sqrt[3]{4 p}}^{2}})$

Step 7

Combine and simplify the denominator.

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Step 7.1

Multiply $\frac{\sqrt[3]{9}}{\sqrt[3]{4 p}}$ by $\frac{{\sqrt[3]{4 p}}^{2}}{{\sqrt[3]{4 p}}^{2}}$ .

$\frac{1}{p} \cdot \frac{\sqrt[3]{9} {\sqrt[3]{4 p}}^{2}}{\sqrt[3]{4 p} {\sqrt[3]{4 p}}^{2}}$

Step 7.2

Raise $\sqrt[3]{4 p}$ to the power of $1$ .

$\frac{1}{p} \cdot \frac{\sqrt[3]{9} {\sqrt[3]{4 p}}^{2}}{{\sqrt[3]{4 p}}^{1} {\sqrt[3]{4 p}}^{2}}$

Step 7.3

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

$\frac{1}{p} \cdot \frac{\sqrt[3]{9} {\sqrt[3]{4 p}}^{2}}{{\sqrt[3]{4 p}}^{1 + 2}}$

Step 7.4

Add $1$ and $2$ .

$\frac{1}{p} \cdot \frac{\sqrt[3]{9} {\sqrt[3]{4 p}}^{2}}{{\sqrt[3]{4 p}}^{3}}$

Step 7.5

Rewrite ${\sqrt[3]{4 p}}^{3}$ as $4 p$ .

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Step 7.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{4 p}$ as ${(4 p)}^{\frac{1}{3}}$ .

$\frac{1}{p} \cdot \frac{\sqrt[3]{9} {\sqrt[3]{4 p}}^{2}}{{({(4 p)}^{\frac{1}{3}})}^{3}}$

Step 7.5.2

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

$\frac{1}{p} \cdot \frac{\sqrt[3]{9} {\sqrt[3]{4 p}}^{2}}{{(4 p)}^{\frac{1}{3} \cdot 3}}$

Step 7.5.3

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

$\frac{1}{p} \cdot \frac{\sqrt[3]{9} {\sqrt[3]{4 p}}^{2}}{{(4 p)}^{\frac{3}{3}}}$

Step 7.5.4

Cancel the common factor of $3$ .

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Step 7.5.4.1

Cancel the common factor.

$\frac{1}{p} \cdot \frac{\sqrt[3]{9} {\sqrt[3]{4 p}}^{2}}{{(4 p)}^{\frac{3}{3}}}$

Step 7.5.4.2

Rewrite the expression.

$\frac{1}{p} \cdot \frac{\sqrt[3]{9} {\sqrt[3]{4 p}}^{2}}{{(4 p)}^{1}}$

Step 7.5.5

Simplify.

$\frac{1}{p} \cdot \frac{\sqrt[3]{9} {\sqrt[3]{4 p}}^{2}}{4 p}$

Step 8

Combine.

$\frac{1 (\sqrt[3]{9} {\sqrt[3]{4 p}}^{2})}{p (4 p)}$

Step 9

Multiply $p$ by $p$ by adding the exponents.

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Step 9.1

Move $p$ .

$\frac{1 (\sqrt[3]{9} {\sqrt[3]{4 p}}^{2})}{p \cdot p \cdot 4}$

Step 9.2

Multiply $p$ by $p$ .

$\frac{1 (\sqrt[3]{9} {\sqrt[3]{4 p}}^{2})}{p^{2} \cdot 4}$

Step 10

Multiply $\sqrt[3]{9} {\sqrt[3]{4 p}}^{2}$ by $1$ .

$\frac{\sqrt[3]{9} {\sqrt[3]{4 p}}^{2}}{p^{2} \cdot 4}$

Step 11

Simplify the numerator.

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Step 11.1

Rewrite ${\sqrt[3]{4 p}}^{2}$ as $\sqrt[3]{{(4 p)}^{2}}$ .

$\frac{\sqrt[3]{9} \sqrt[3]{{(4 p)}^{2}}}{p^{2} \cdot 4}$

Step 11.2

Apply the product rule to $4 p$ .

$\frac{\sqrt[3]{9} \sqrt[3]{4^{2} p^{2}}}{p^{2} \cdot 4}$

Step 11.3

Raise $4$ to the power of $2$ .

$\frac{\sqrt[3]{9} \sqrt[3]{16 p^{2}}}{p^{2} \cdot 4}$

Step 11.4

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

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Step 11.4.1

Factor $8$ out of $16$ .

$\frac{\sqrt[3]{9} \sqrt[3]{8 (2) p^{2}}}{p^{2} \cdot 4}$

Step 11.4.2

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

$\frac{\sqrt[3]{9} \sqrt[3]{2^{3} \cdot 2 p^{2}}}{p^{2} \cdot 4}$

Step 11.4.3

Add parentheses.

$\frac{\sqrt[3]{9} \sqrt[3]{2^{3} \cdot (2 p^{2})}}{p^{2} \cdot 4}$

Step 11.5

Pull terms out from under the radical.

$\frac{\sqrt[3]{9} \cdot 2 \sqrt[3]{2 p^{2}}}{p^{2} \cdot 4}$

Step 11.6

Combine exponents.

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Step 11.6.1

Combine using the product rule for radicals.

$\frac{2 \sqrt[3]{9 (2 p^{2})}}{p^{2} \cdot 4}$

Step 11.6.2

Multiply $9$ by $2$ .

$\frac{2 \sqrt[3]{18 p^{2}}}{p^{2} \cdot 4}$

Step 12

Reduce the expression by cancelling the common factors.

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Step 12.1

Cancel the common factor of $2$ and $4$ .

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Step 12.1.1

Factor $2$ out of $2 \sqrt[3]{18 p^{2}}$ .

$\frac{2 (\sqrt[3]{18 p^{2}})}{p^{2} \cdot 4}$

Step 12.1.2

Cancel the common factors.

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Step 12.1.2.1

Factor $2$ out of $p^{2} \cdot 4$ .

$\frac{2 (\sqrt[3]{18 p^{2}})}{2 (p^{2} \cdot 2)}$

Step 12.1.2.2

Cancel the common factor.

$\frac{2 \sqrt[3]{18 p^{2}}}{2 (p^{2} \cdot 2)}$

Step 12.1.2.3

Rewrite the expression.

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

Step 12.2

Move $2$ to the left of $p^{2}$ .

$\frac{\sqrt[3]{18 p^{2}}}{2 p^{2}}$