Basic Math Examples

$\frac{\frac{b}{b + 2} + \frac{1}{3 b}}{\frac{b}{6 b + 12} + \frac{2}{b}}$

Step 1

Multiply the numerator and denominator of the fraction by $(b + 2) \cdot 3 b (6 b + 12)$ .

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Step 1.1

Multiply $\frac{\frac{b}{b + 2} + \frac{1}{3 b}}{\frac{b}{6 b + 12} + \frac{2}{b}}$ by $\frac{(b + 2) \cdot 3 b (6 b + 12)}{(b + 2) \cdot 3 b (6 b + 12)}$ .

$\frac{(b + 2) \cdot 3 b (6 b + 12)}{(b + 2) \cdot 3 b (6 b + 12)} \cdot \frac{\frac{b}{b + 2} + \frac{1}{3 b}}{\frac{b}{6 b + 12} + \frac{2}{b}}$

Step 1.2

Combine.

$\frac{(b + 2) \cdot 3 b (6 b + 12) (\frac{b}{b + 2} + \frac{1}{3 b})}{(b + 2) \cdot 3 b (6 b + 12) (\frac{b}{6 b + 12} + \frac{2}{b})}$

Step 2

Apply the distributive property.

$\frac{(b + 2) \cdot 3 b (6 b + 12) \frac{b}{b + 2} + (b + 2) \cdot 3 b (6 b + 12) \frac{1}{3 b}}{(b + 2) \cdot 3 b (6 b + 12) \frac{b}{6 b + 12} + (b + 2) \cdot 3 b (6 b + 12) \frac{2}{b}}$

Step 3

Simplify by cancelling.

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Step 3.1

Cancel the common factor of $b + 2$ .

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Step 3.1.1

Factor $b + 2$ out of $(b + 2) \cdot 3 b (6 b + 12)$ .

$\frac{(b + 2) (3 b (6 b + 12)) \frac{b}{b + 2} + (b + 2) \cdot 3 b (6 b + 12) \frac{1}{3 b}}{(b + 2) \cdot 3 b (6 b + 12) \frac{b}{6 b + 12} + (b + 2) \cdot 3 b (6 b + 12) \frac{2}{b}}$

Step 3.1.2

Cancel the common factor.

$\frac{(b + 2) (3 b (6 b + 12)) \frac{b}{b + 2} + (b + 2) \cdot 3 b (6 b + 12) \frac{1}{3 b}}{(b + 2) \cdot 3 b (6 b + 12) \frac{b}{6 b + 12} + (b + 2) \cdot 3 b (6 b + 12) \frac{2}{b}}$

Step 3.1.3

Rewrite the expression.

$\frac{3 b (6 b + 12) b + (b + 2) \cdot 3 b (6 b + 12) \frac{1}{3 b}}{(b + 2) \cdot 3 b (6 b + 12) \frac{b}{6 b + 12} + (b + 2) \cdot 3 b (6 b + 12) \frac{2}{b}}$

Step 3.2

Raise $b$ to the power of $1$ .

$\frac{3 (b^{1} b) (6 b + 12) + (b + 2) \cdot 3 b (6 b + 12) \frac{1}{3 b}}{(b + 2) \cdot 3 b (6 b + 12) \frac{b}{6 b + 12} + (b + 2) \cdot 3 b (6 b + 12) \frac{2}{b}}$

Step 3.3

Raise $b$ to the power of $1$ .

$\frac{3 (b^{1} b^{1}) (6 b + 12) + (b + 2) \cdot 3 b (6 b + 12) \frac{1}{3 b}}{(b + 2) \cdot 3 b (6 b + 12) \frac{b}{6 b + 12} + (b + 2) \cdot 3 b (6 b + 12) \frac{2}{b}}$

Step 3.4

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

$\frac{3 b^{1 + 1} (6 b + 12) + (b + 2) \cdot 3 b (6 b + 12) \frac{1}{3 b}}{(b + 2) \cdot 3 b (6 b + 12) \frac{b}{6 b + 12} + (b + 2) \cdot 3 b (6 b + 12) \frac{2}{b}}$

Step 3.5

Add $1$ and $1$ .

$\frac{3 b^{2} (6 b + 12) + (b + 2) \cdot 3 b (6 b + 12) \frac{1}{3 b}}{(b + 2) \cdot 3 b (6 b + 12) \frac{b}{6 b + 12} + (b + 2) \cdot 3 b (6 b + 12) \frac{2}{b}}$

Step 3.6

Cancel the common factor of $3 b$ .

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Step 3.6.1

Factor $3 b$ out of $(b + 2) \cdot 3 b (6 b + 12)$ .

$\frac{3 b^{2} (6 b + 12) + 3 b ((b + 2) (6 b + 12)) \frac{1}{3 b}}{(b + 2) \cdot 3 b (6 b + 12) \frac{b}{6 b + 12} + (b + 2) \cdot 3 b (6 b + 12) \frac{2}{b}}$

Step 3.6.2

Cancel the common factor.

$\frac{3 b^{2} (6 b + 12) + 3 b ((b + 2) (6 b + 12)) \frac{1}{3 b}}{(b + 2) \cdot 3 b (6 b + 12) \frac{b}{6 b + 12} + (b + 2) \cdot 3 b (6 b + 12) \frac{2}{b}}$

Step 3.6.3

Rewrite the expression.

$\frac{3 b^{2} (6 b + 12) + (b + 2) (6 b + 12)}{(b + 2) \cdot 3 b (6 b + 12) \frac{b}{6 b + 12} + (b + 2) \cdot 3 b (6 b + 12) \frac{2}{b}}$

Step 3.7

Cancel the common factor of $6 b + 12$ .

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Step 3.7.1

Factor $6 b + 12$ out of $(b + 2) \cdot 3 b (6 b + 12)$ .

$\frac{3 b^{2} (6 b + 12) + (b + 2) (6 b + 12)}{(6 b + 12) ((b + 2) \cdot 3 b) \frac{b}{6 b + 12} + (b + 2) \cdot 3 b (6 b + 12) \frac{2}{b}}$

Step 3.7.2

Cancel the common factor.

$\frac{3 b^{2} (6 b + 12) + (b + 2) (6 b + 12)}{(6 b + 12) ((b + 2) \cdot 3 b) \frac{b}{6 b + 12} + (b + 2) \cdot 3 b (6 b + 12) \frac{2}{b}}$

Step 3.7.3

Rewrite the expression.

$\frac{3 b^{2} (6 b + 12) + (b + 2) (6 b + 12)}{(b + 2) \cdot 3 b \cdot b + (b + 2) \cdot 3 b (6 b + 12) \frac{2}{b}}$

Step 3.8

Raise $b$ to the power of $1$ .

$\frac{3 b^{2} (6 b + 12) + (b + 2) (6 b + 12)}{(b + 2) \cdot 3 (b^{1} b) + (b + 2) \cdot 3 b (6 b + 12) \frac{2}{b}}$

Step 3.9

Raise $b$ to the power of $1$ .

$\frac{3 b^{2} (6 b + 12) + (b + 2) (6 b + 12)}{(b + 2) \cdot 3 (b^{1} b^{1}) + (b + 2) \cdot 3 b (6 b + 12) \frac{2}{b}}$

Step 3.10

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

$\frac{3 b^{2} (6 b + 12) + (b + 2) (6 b + 12)}{(b + 2) \cdot 3 b^{1 + 1} + (b + 2) \cdot 3 b (6 b + 12) \frac{2}{b}}$

Step 3.11

Add $1$ and $1$ .

$\frac{3 b^{2} (6 b + 12) + (b + 2) (6 b + 12)}{(b + 2) \cdot 3 b^{2} + (b + 2) \cdot 3 b (6 b + 12) \frac{2}{b}}$

Step 3.12

Cancel the common factor of $b$ .

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Step 3.12.1

Factor $b$ out of $(b + 2) \cdot 3 b (6 b + 12)$ .

$\frac{3 b^{2} (6 b + 12) + (b + 2) (6 b + 12)}{(b + 2) \cdot 3 b^{2} + b ((b + 2) \cdot 3 (6 b + 12)) \frac{2}{b}}$

Step 3.12.2

Cancel the common factor.

$\frac{3 b^{2} (6 b + 12) + (b + 2) (6 b + 12)}{(b + 2) \cdot 3 b^{2} + b ((b + 2) \cdot 3 (6 b + 12)) \frac{2}{b}}$

Step 3.12.3

Rewrite the expression.

$\frac{3 b^{2} (6 b + 12) + (b + 2) (6 b + 12)}{(b + 2) \cdot 3 b^{2} + (b + 2) \cdot 3 (6 b + 12) \cdot 2}$

Step 3.13

Multiply $2$ by $3$ .

$\frac{3 b^{2} (6 b + 12) + (b + 2) (6 b + 12)}{(b + 2) \cdot 3 b^{2} + (b + 2) \cdot 6 (6 b + 12)}$

Step 4

Simplify the numerator.

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Step 4.1

Factor $6 b + 12$ out of $3 b^{2} (6 b + 12) + (b + 2) (6 b + 12)$ .

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Step 4.1.1

Factor $6 b + 12$ out of $3 b^{2} (6 b + 12)$ .

$\frac{(6 b + 12) (3 b^{2}) + (b + 2) (6 b + 12)}{(b + 2) \cdot 3 b^{2} + (b + 2) \cdot 6 (6 b + 12)}$

Step 4.1.2

Factor $6 b + 12$ out of $(b + 2) (6 b + 12)$ .

$\frac{(6 b + 12) (3 b^{2}) + (6 b + 12) (b + 2)}{(b + 2) \cdot 3 b^{2} + (b + 2) \cdot 6 (6 b + 12)}$

Step 4.1.3

Factor $6 b + 12$ out of $(6 b + 12) (3 b^{2}) + (6 b + 12) (b + 2)$ .

$\frac{(6 b + 12) (3 b^{2} + b + 2)}{(b + 2) \cdot 3 b^{2} + (b + 2) \cdot 6 (6 b + 12)}$

Step 4.2

Factor $6$ out of $6 b + 12$ .

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Step 4.2.1

Factor $6$ out of $6 b$ .

$\frac{(6 (b) + 12) (3 b^{2} + b + 2)}{(b + 2) \cdot 3 b^{2} + (b + 2) \cdot 6 (6 b + 12)}$

Step 4.2.2

Factor $6$ out of $12$ .

$\frac{(6 b + 6 \cdot 2) (3 b^{2} + b + 2)}{(b + 2) \cdot 3 b^{2} + (b + 2) \cdot 6 (6 b + 12)}$

Step 4.2.3

Factor $6$ out of $6 b + 6 \cdot 2$ .

$\frac{6 (b + 2) (3 b^{2} + b + 2)}{(b + 2) \cdot 3 b^{2} + (b + 2) \cdot 6 (6 b + 12)}$

Step 5

Simplify the denominator.

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Step 5.1

Factor $(b + 2) \cdot 3$ out of $(b + 2) \cdot 3 b^{2} + (b + 2) \cdot 6 (6 b + 12)$ .

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Step 5.1.1

Factor $(b + 2) \cdot 3$ out of $(b + 2) \cdot 3 b^{2}$ .

$\frac{6 (b + 2) (3 b^{2} + b + 2)}{(b + 2) \cdot 3 (b^{2}) + (b + 2) \cdot 6 (6 b + 12)}$

Step 5.1.2

Factor $(b + 2) \cdot 3$ out of $(b + 2) \cdot 6 (6 b + 12)$ .

$\frac{6 (b + 2) (3 b^{2} + b + 2)}{(b + 2) \cdot 3 (b^{2}) + (b + 2) \cdot 3 (2 (6 b + 12))}$

Step 5.1.3

Factor $(b + 2) \cdot 3$ out of $(b + 2) \cdot 3 (b^{2}) + (b + 2) \cdot 3 (2 (6 b + 12))$ .

$\frac{6 (b + 2) (3 b^{2} + b + 2)}{(b + 2) \cdot 3 (b^{2} + 2 (6 b + 12))}$

Step 5.2

Apply the distributive property.

$\frac{6 (b + 2) (3 b^{2} + b + 2)}{(b + 2) \cdot 3 (b^{2} + 2 (6 b) + 2 \cdot 12)}$

Step 5.3

Multiply $6$ by $2$ .

$\frac{6 (b + 2) (3 b^{2} + b + 2)}{(b + 2) \cdot 3 (b^{2} + 12 b + 2 \cdot 12)}$

Step 5.4

Multiply $2$ by $12$ .

$\frac{6 (b + 2) (3 b^{2} + b + 2)}{(b + 2) \cdot 3 (b^{2} + 12 b + 24)}$

Step 6

Reduce the expression by cancelling the common factors.

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Step 6.1

Cancel the common factor of $6$ and $3$ .

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Step 6.1.1

Factor $3$ out of $6 (b + 2) (3 b^{2} + b + 2)$ .

$\frac{3 (2 (b + 2) (3 b^{2} + b + 2))}{(b + 2) \cdot 3 (b^{2} + 12 b + 24)}$

Step 6.1.2

Cancel the common factors.

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Step 6.1.2.1

Factor $3$ out of $(b + 2) \cdot 3 (b^{2} + 12 b + 24)$ .

$\frac{3 (2 (b + 2) (3 b^{2} + b + 2))}{3 ((b + 2) (b^{2} + 12 b + 24))}$

Step 6.1.2.2

Cancel the common factor.

$\frac{3 (2 (b + 2) (3 b^{2} + b + 2))}{3 ((b + 2) (b^{2} + 12 b + 24))}$

Step 6.1.2.3

Rewrite the expression.

$\frac{2 (b + 2) (3 b^{2} + b + 2)}{(b + 2) (b^{2} + 12 b + 24)}$

Step 6.2

Cancel the common factor of $b + 2$ .

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Step 6.2.1

Cancel the common factor.

$\frac{2 (b + 2) (3 b^{2} + b + 2)}{(b + 2) (b^{2} + 12 b + 24)}$

Step 6.2.2

Rewrite the expression.

$\frac{2 (3 b^{2} + b + 2)}{b^{2} + 12 b + 24}$