Pre-Algebra Examples

$\frac{3 a^{2} b^{3}}{3 \sqrt{125 a^{3} b^{3}}}$

Step 1

Cancel the common factor of $3$ .

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Step 1.1

Cancel the common factor.

$\frac{3 a^{2} b^{3}}{3 \sqrt{125 a^{3} b^{3}}}$

Step 1.2

Rewrite the expression.

$\frac{a^{2} b^{3}}{\sqrt{125 a^{3} b^{3}}}$

Step 2

Simplify the denominator.

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Step 2.1

Rewrite $125 a^{3} b^{3}$ as ${(5 a b)}^{2} \cdot (5 a b)$ .

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Step 2.1.1

Factor $25$ out of $125$ .

$\frac{a^{2} b^{3}}{\sqrt{25 (5) a^{3} b^{3}}}$

Step 2.1.2

Rewrite $25$ as $5^{2}$ .

$\frac{a^{2} b^{3}}{\sqrt{5^{2} \cdot 5 a^{3} b^{3}}}$

Step 2.1.3

Factor out $a^{2}$ .

$\frac{a^{2} b^{3}}{\sqrt{5^{2} \cdot 5 (a^{2} a) b^{3}}}$

Step 2.1.4

Factor out $b^{2}$ .

$\frac{a^{2} b^{3}}{\sqrt{5^{2} \cdot 5 (a^{2} a) (b^{2} b)}}$

Step 2.1.5

Move $a$ .

$\frac{a^{2} b^{3}}{\sqrt{5^{2} \cdot 5 (a^{2}) b^{2} a b}}$

Step 2.1.6

Move $5$ .

$\frac{a^{2} b^{3}}{\sqrt{(5^{2} (a^{2})) b^{2} \cdot 5 a b}}$

Step 2.1.7

Rewrite $(5^{2} (a^{2})) b^{2}$ as ${(5 a b)}^{2}$ .

$\frac{a^{2} b^{3}}{\sqrt{{(5 a b)}^{2} \cdot 5 a b}}$

Step 2.1.8

Add parentheses.

$\frac{a^{2} b^{3}}{\sqrt{{(5 a b)}^{2} \cdot 5 (a b)}}$

Step 2.1.9

Add parentheses.

$\frac{a^{2} b^{3}}{\sqrt{{(5 a b)}^{2} \cdot (5 a b)}}$

Step 2.2

Pull terms out from under the radical.

$\frac{a^{2} b^{3}}{5 a b \sqrt{5 a b}}$

Step 3

Reduce the expression by cancelling the common factors.

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Step 3.1

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

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Step 3.1.1

Factor $a$ out of $a^{2} b^{3}$ .

$\frac{a (a b^{3})}{5 a b \sqrt{5 a b}}$

Step 3.1.2

Cancel the common factors.

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Step 3.1.2.1

Factor $a$ out of $5 a b \sqrt{5 a b}$ .

$\frac{a (a b^{3})}{a (5 b \sqrt{5 a b})}$

Step 3.1.2.2

Cancel the common factor.

$\frac{a (a b^{3})}{a (5 b \sqrt{5 a b})}$

Step 3.1.2.3

Rewrite the expression.

$\frac{a b^{3}}{5 b \sqrt{5 a b}}$

Step 3.2

Cancel the common factor of $b^{3}$ and $b$ .

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Step 3.2.1

Factor $b$ out of $a b^{3}$ .

$\frac{b (a b^{2})}{5 b \sqrt{5 a b}}$

Step 3.2.2

Cancel the common factors.

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Step 3.2.2.1

Factor $b$ out of $5 b \sqrt{5 a b}$ .

$\frac{b (a b^{2})}{b (5 \sqrt{5 a b})}$

Step 3.2.2.2

Cancel the common factor.

$\frac{b (a b^{2})}{b (5 \sqrt{5 a b})}$

Step 3.2.2.3

Rewrite the expression.

$\frac{a b^{2}}{5 \sqrt{5 a b}}$

Step 4

Multiply $\frac{a b^{2}}{5 \sqrt{5 a b}}$ by $\frac{\sqrt{5 a b}}{\sqrt{5 a b}}$ .

$\frac{a b^{2}}{5 \sqrt{5 a b}} \cdot \frac{\sqrt{5 a b}}{\sqrt{5 a b}}$

Step 5

Simplify terms.

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Step 5.1

Combine and simplify the denominator.

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Step 5.1.1

Multiply $\frac{a b^{2}}{5 \sqrt{5 a b}}$ by $\frac{\sqrt{5 a b}}{\sqrt{5 a b}}$ .

$\frac{a b^{2} \sqrt{5 a b}}{5 \sqrt{5 a b} \sqrt{5 a b}}$

Step 5.1.2

Move $\sqrt{5 a b}$ .

$\frac{a b^{2} \sqrt{5 a b}}{5 (\sqrt{5 a b} \sqrt{5 a b})}$

Step 5.1.3

Raise $\sqrt{5 a b}$ to the power of $1$ .

$\frac{a b^{2} \sqrt{5 a b}}{5 ({\sqrt{5 a b}}^{1} \sqrt{5 a b})}$

Step 5.1.4

Raise $\sqrt{5 a b}$ to the power of $1$ .

$\frac{a b^{2} \sqrt{5 a b}}{5 ({\sqrt{5 a b}}^{1} {\sqrt{5 a b}}^{1})}$

Step 5.1.5

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

$\frac{a b^{2} \sqrt{5 a b}}{5 {\sqrt{5 a b}}^{1 + 1}}$

Step 5.1.6

Add $1$ and $1$ .

$\frac{a b^{2} \sqrt{5 a b}}{5 {\sqrt{5 a b}}^{2}}$

Step 5.1.7

Rewrite ${\sqrt{5 a b}}^{2}$ as $5 a b$ .

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Step 5.1.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5 a b}$ as ${(5 a b)}^{\frac{1}{2}}$ .

$\frac{a b^{2} \sqrt{5 a b}}{5 {({(5 a b)}^{\frac{1}{2}})}^{2}}$

Step 5.1.7.2

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

$\frac{a b^{2} \sqrt{5 a b}}{5 {(5 a b)}^{\frac{1}{2} \cdot 2}}$

Step 5.1.7.3

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

$\frac{a b^{2} \sqrt{5 a b}}{5 {(5 a b)}^{\frac{2}{2}}}$

Step 5.1.7.4

Cancel the common factor of $2$ .

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Step 5.1.7.4.1

Cancel the common factor.

$\frac{a b^{2} \sqrt{5 a b}}{5 {(5 a b)}^{\frac{2}{2}}}$

Step 5.1.7.4.2

Rewrite the expression.

$\frac{a b^{2} \sqrt{5 a b}}{5 {(5 a b)}^{1}}$

Step 5.1.7.5

Simplify.

$\frac{a b^{2} \sqrt{5 a b}}{5 (5 a b)}$

Step 5.2

Cancel the common factor of $a$ .

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Step 5.2.1

Cancel the common factor.

$\frac{a b^{2} \sqrt{5 a b}}{5 (5 a b)}$

Step 5.2.2

Rewrite the expression.

$\frac{b^{2} \sqrt{5 a b}}{5 (5 b)}$

Step 5.3

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

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Step 5.3.1

Factor $b$ out of $b^{2} \sqrt{5 a b}$ .

$\frac{b (b \sqrt{5 a b})}{5 (5 b)}$

Step 5.3.2

Cancel the common factors.

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Step 5.3.2.1

Factor $b$ out of $5 (5 b)$ .

$\frac{b (b \sqrt{5 a b})}{b (5 \cdot (5))}$

Step 5.3.2.2

Cancel the common factor.

$\frac{b (b \sqrt{5 a b})}{b (5 \cdot (5))}$

Step 5.3.2.3

Rewrite the expression.

$\frac{b \sqrt{5 a b}}{5 \cdot (5)}$

Step 6

Multiply $5$ by $5$ .

$\frac{b \sqrt{5 a b}}{25}$