Pre-Algebra Examples

$\sqrt[3]{\frac{25}{4 a^{2}}}$

Step 1

Rewrite $\sqrt[3]{\frac{25}{4 a^{2}}}$ as $\frac{\sqrt[3]{25}}{\sqrt[3]{4 a^{2}}}$ .

$\frac{\sqrt[3]{25}}{\sqrt[3]{4 a^{2}}}$

Step 2

Multiply $\frac{\sqrt[3]{25}}{\sqrt[3]{4 a^{2}}}$ by $\frac{{\sqrt[3]{4 a^{2}}}^{2}}{{\sqrt[3]{4 a^{2}}}^{2}}$ .

$\frac{\sqrt[3]{25}}{\sqrt[3]{4 a^{2}}} \cdot \frac{{\sqrt[3]{4 a^{2}}}^{2}}{{\sqrt[3]{4 a^{2}}}^{2}}$

Step 3

Combine and simplify the denominator.

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Step 3.1

Multiply $\frac{\sqrt[3]{25}}{\sqrt[3]{4 a^{2}}}$ by $\frac{{\sqrt[3]{4 a^{2}}}^{2}}{{\sqrt[3]{4 a^{2}}}^{2}}$ .

$\frac{\sqrt[3]{25} {\sqrt[3]{4 a^{2}}}^{2}}{\sqrt[3]{4 a^{2}} {\sqrt[3]{4 a^{2}}}^{2}}$

Step 3.2

Raise $\sqrt[3]{4 a^{2}}$ to the power of $1$ .

$\frac{\sqrt[3]{25} {\sqrt[3]{4 a^{2}}}^{2}}{{\sqrt[3]{4 a^{2}}}^{1} {\sqrt[3]{4 a^{2}}}^{2}}$

Step 3.3

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

$\frac{\sqrt[3]{25} {\sqrt[3]{4 a^{2}}}^{2}}{{\sqrt[3]{4 a^{2}}}^{1 + 2}}$

Step 3.4

Add $1$ and $2$ .

$\frac{\sqrt[3]{25} {\sqrt[3]{4 a^{2}}}^{2}}{{\sqrt[3]{4 a^{2}}}^{3}}$

Step 3.5

Rewrite ${\sqrt[3]{4 a^{2}}}^{3}$ as $4 a^{2}$ .

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Step 3.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{4 a^{2}}$ as ${(4 a^{2})}^{\frac{1}{3}}$ .

$\frac{\sqrt[3]{25} {\sqrt[3]{4 a^{2}}}^{2}}{{({(4 a^{2})}^{\frac{1}{3}})}^{3}}$

Step 3.5.2

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

$\frac{\sqrt[3]{25} {\sqrt[3]{4 a^{2}}}^{2}}{{(4 a^{2})}^{\frac{1}{3} \cdot 3}}$

Step 3.5.3

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

$\frac{\sqrt[3]{25} {\sqrt[3]{4 a^{2}}}^{2}}{{(4 a^{2})}^{\frac{3}{3}}}$

Step 3.5.4

Cancel the common factor of $3$ .

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Step 3.5.4.1

Cancel the common factor.

$\frac{\sqrt[3]{25} {\sqrt[3]{4 a^{2}}}^{2}}{{(4 a^{2})}^{\frac{3}{3}}}$

Step 3.5.4.2

Rewrite the expression.

$\frac{\sqrt[3]{25} {\sqrt[3]{4 a^{2}}}^{2}}{{(4 a^{2})}^{1}}$

Step 3.5.5

Simplify.

$\frac{\sqrt[3]{25} {\sqrt[3]{4 a^{2}}}^{2}}{4 a^{2}}$

Step 4

Simplify the numerator.

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Step 4.1

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

$\frac{\sqrt[3]{25} \sqrt[3]{{(4 a^{2})}^{2}}}{4 a^{2}}$

Step 4.2

Apply the product rule to $4 a^{2}$ .

$\frac{\sqrt[3]{25} \sqrt[3]{4^{2} {(a^{2})}^{2}}}{4 a^{2}}$

Step 4.3

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

$\frac{\sqrt[3]{25} \sqrt[3]{16 {(a^{2})}^{2}}}{4 a^{2}}$

Step 4.4

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

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Step 4.4.1

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

$\frac{\sqrt[3]{25} \sqrt[3]{16 a^{2 \cdot 2}}}{4 a^{2}}$

Step 4.4.2

Multiply $2$ by $2$ .

$\frac{\sqrt[3]{25} \sqrt[3]{16 a^{4}}}{4 a^{2}}$

Step 4.5

Rewrite $16 a^{4}$ as ${(2 a)}^{3} \cdot (2 a)$ .

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Step 4.5.1

Factor $8$ out of $16$ .

$\frac{\sqrt[3]{25} \sqrt[3]{8 (2) a^{4}}}{4 a^{2}}$

Step 4.5.2

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

$\frac{\sqrt[3]{25} \sqrt[3]{2^{3} \cdot 2 a^{4}}}{4 a^{2}}$

Step 4.5.3

Factor out $a^{3}$ .

$\frac{\sqrt[3]{25} \sqrt[3]{2^{3} \cdot 2 (a^{3} a)}}{4 a^{2}}$

Step 4.5.4

Move $2$ .

$\frac{\sqrt[3]{25} \sqrt[3]{2^{3} a^{3} \cdot 2 a}}{4 a^{2}}$

Step 4.5.5

Rewrite $2^{3} a^{3}$ as ${(2 a)}^{3}$ .

$\frac{\sqrt[3]{25} \sqrt[3]{{(2 a)}^{3} \cdot 2 a}}{4 a^{2}}$

Step 4.5.6

Add parentheses.

$\frac{\sqrt[3]{25} \sqrt[3]{{(2 a)}^{3} \cdot (2 a)}}{4 a^{2}}$

Step 4.6

Pull terms out from under the radical.

$\frac{\sqrt[3]{25} \cdot 2 a \sqrt[3]{2 a}}{4 a^{2}}$

Step 4.7

Combine exponents.

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Step 4.7.1

Combine using the product rule for radicals.

$\frac{2 a \sqrt[3]{25 (2 a)}}{4 a^{2}}$

Step 4.7.2

Multiply $25$ by $2$ .

$\frac{2 a \sqrt[3]{50 a}}{4 a^{2}}$

Step 5

Reduce the expression by cancelling the common factors.

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Step 5.1

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

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Step 5.1.1

Factor $2$ out of $2 a \sqrt[3]{50 a}$ .

$\frac{2 (a \sqrt[3]{50 a})}{4 a^{2}}$

Step 5.1.2

Cancel the common factors.

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Step 5.1.2.1

Factor $2$ out of $4 a^{2}$ .

$\frac{2 (a \sqrt[3]{50 a})}{2 (2 a^{2})}$

Step 5.1.2.2

Cancel the common factor.

$\frac{2 (a \sqrt[3]{50 a})}{2 (2 a^{2})}$

Step 5.1.2.3

Rewrite the expression.

$\frac{a \sqrt[3]{50 a}}{2 a^{2}}$

Step 5.2

Cancel the common factor of $a$ and $a^{2}$ .

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Step 5.2.1

Factor $a$ out of $a \sqrt[3]{50 a}$ .

$\frac{a (\sqrt[3]{50 a})}{2 a^{2}}$

Step 5.2.2

Cancel the common factors.

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Step 5.2.2.1

Factor $a$ out of $2 a^{2}$ .

$\frac{a (\sqrt[3]{50 a})}{a (2 a)}$

Step 5.2.2.2

Cancel the common factor.

$\frac{a \sqrt[3]{50 a}}{a (2 a)}$

Step 5.2.2.3

Rewrite the expression.

$\frac{\sqrt[3]{50 a}}{2 a}$