Algebra Examples

$\frac{\frac{3}{2} a^{2} - 2 a b + \frac{2}{3} b^{2}}{\frac{1}{4} a^{2} - \frac{1}{9} b^{2}} + \frac{6 b}{\frac{3}{4} a + \frac{1}{2} b}$

Step 1

Simplify each term.

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Step 1.1

Simplify the numerator.

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Step 1.1.1

Combine $\frac{3}{2}$ and $a^{2}$ .

$\frac{\frac{3 a^{2}}{2} - 2 a b + \frac{2}{3} b^{2}}{\frac{1}{4} a^{2} - \frac{1}{9} b^{2}} + \frac{6 b}{\frac{3}{4} a + \frac{1}{2} b}$

Step 1.1.2

Combine $\frac{2}{3}$ and $b^{2}$ .

$\frac{\frac{3 a^{2}}{2} - 2 a b + \frac{2 b^{2}}{3}}{\frac{1}{4} a^{2} - \frac{1}{9} b^{2}} + \frac{6 b}{\frac{3}{4} a + \frac{1}{2} b}$

Step 1.1.3

To write $- 2 a b$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{\frac{3 a^{2}}{2} - 2 a b \cdot \frac{2}{2} + \frac{2 b^{2}}{3}}{\frac{1}{4} a^{2} - \frac{1}{9} b^{2}} + \frac{6 b}{\frac{3}{4} a + \frac{1}{2} b}$

Step 1.1.4

Combine $- 2 a b$ and $\frac{2}{2}$ .

$\frac{\frac{3 a^{2}}{2} + \frac{- 2 a b \cdot 2}{2} + \frac{2 b^{2}}{3}}{\frac{1}{4} a^{2} - \frac{1}{9} b^{2}} + \frac{6 b}{\frac{3}{4} a + \frac{1}{2} b}$

Step 1.1.5

Combine the numerators over the common denominator.

$\frac{\frac{3 a^{2} - 2 a b \cdot 2}{2} + \frac{2 b^{2}}{3}}{\frac{1}{4} a^{2} - \frac{1}{9} b^{2}} + \frac{6 b}{\frac{3}{4} a + \frac{1}{2} b}$

Step 1.1.6

To write $\frac{3 a^{2} - 2 a b \cdot 2}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{\frac{3 a^{2} - 2 a b \cdot 2}{2} \cdot \frac{3}{3} + \frac{2 b^{2}}{3}}{\frac{1}{4} a^{2} - \frac{1}{9} b^{2}} + \frac{6 b}{\frac{3}{4} a + \frac{1}{2} b}$

Step 1.1.7

To write $\frac{2 b^{2}}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{\frac{3 a^{2} - 2 a b \cdot 2}{2} \cdot \frac{3}{3} + \frac{2 b^{2}}{3} \cdot \frac{2}{2}}{\frac{1}{4} a^{2} - \frac{1}{9} b^{2}} + \frac{6 b}{\frac{3}{4} a + \frac{1}{2} b}$

Step 1.1.8

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.1.8.1

Multiply $\frac{3 a^{2} - 2 a b \cdot 2}{2}$ by $\frac{3}{3}$ .

$\frac{\frac{(3 a^{2} - 2 a b \cdot 2) \cdot 3}{2 \cdot 3} + \frac{2 b^{2}}{3} \cdot \frac{2}{2}}{\frac{1}{4} a^{2} - \frac{1}{9} b^{2}} + \frac{6 b}{\frac{3}{4} a + \frac{1}{2} b}$

Step 1.1.8.2

Multiply $2$ by $3$ .

$\frac{\frac{(3 a^{2} - 2 a b \cdot 2) \cdot 3}{6} + \frac{2 b^{2}}{3} \cdot \frac{2}{2}}{\frac{1}{4} a^{2} - \frac{1}{9} b^{2}} + \frac{6 b}{\frac{3}{4} a + \frac{1}{2} b}$

Step 1.1.8.3

Multiply $\frac{2 b^{2}}{3}$ by $\frac{2}{2}$ .

$\frac{\frac{(3 a^{2} - 2 a b \cdot 2) \cdot 3}{6} + \frac{2 b^{2} \cdot 2}{3 \cdot 2}}{\frac{1}{4} a^{2} - \frac{1}{9} b^{2}} + \frac{6 b}{\frac{3}{4} a + \frac{1}{2} b}$

Step 1.1.8.4

Multiply $3$ by $2$ .

$\frac{\frac{(3 a^{2} - 2 a b \cdot 2) \cdot 3}{6} + \frac{2 b^{2} \cdot 2}{6}}{\frac{1}{4} a^{2} - \frac{1}{9} b^{2}} + \frac{6 b}{\frac{3}{4} a + \frac{1}{2} b}$

Step 1.1.9

Combine the numerators over the common denominator.

$\frac{\frac{(3 a^{2} - 2 a b \cdot 2) \cdot 3 + 2 b^{2} \cdot 2}{6}}{\frac{1}{4} a^{2} - \frac{1}{9} b^{2}} + \frac{6 b}{\frac{3}{4} a + \frac{1}{2} b}$

Step 1.1.10

Rewrite $\frac{(3 a^{2} - 2 a b \cdot 2) \cdot 3 + 2 b^{2} \cdot 2}{6}$ in a factored form.

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Step 1.1.10.1

Multiply $2$ by $- 2$ .

$\frac{\frac{(3 a^{2} - 4 a b) \cdot 3 + 2 b^{2} \cdot 2}{6}}{\frac{1}{4} a^{2} - \frac{1}{9} b^{2}} + \frac{6 b}{\frac{3}{4} a + \frac{1}{2} b}$

Step 1.1.10.2

Apply the distributive property.

$\frac{\frac{3 a^{2} \cdot 3 - 4 a b \cdot 3 + 2 b^{2} \cdot 2}{6}}{\frac{1}{4} a^{2} - \frac{1}{9} b^{2}} + \frac{6 b}{\frac{3}{4} a + \frac{1}{2} b}$

Step 1.1.10.3

Multiply $3$ by $3$ .

$\frac{\frac{9 a^{2} - 4 a b \cdot 3 + 2 b^{2} \cdot 2}{6}}{\frac{1}{4} a^{2} - \frac{1}{9} b^{2}} + \frac{6 b}{\frac{3}{4} a + \frac{1}{2} b}$

Step 1.1.10.4

Multiply $3$ by $- 4$ .

$\frac{\frac{9 a^{2} - 12 a b + 2 b^{2} \cdot 2}{6}}{\frac{1}{4} a^{2} - \frac{1}{9} b^{2}} + \frac{6 b}{\frac{3}{4} a + \frac{1}{2} b}$

Step 1.1.10.5

Multiply $2$ by $2$ .

$\frac{\frac{9 a^{2} - 12 a b + 4 b^{2}}{6}}{\frac{1}{4} a^{2} - \frac{1}{9} b^{2}} + \frac{6 b}{\frac{3}{4} a + \frac{1}{2} b}$

Step 1.1.10.6

Factor using the perfect square rule.

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Step 1.1.10.6.1

Rewrite $9 a^{2}$ as ${(3 a)}^{2}$ .

$\frac{\frac{{(3 a)}^{2} - 12 a b + 4 b^{2}}{6}}{\frac{1}{4} a^{2} - \frac{1}{9} b^{2}} + \frac{6 b}{\frac{3}{4} a + \frac{1}{2} b}$

Step 1.1.10.6.2

Rewrite $4 b^{2}$ as ${(2 b)}^{2}$ .

$\frac{\frac{{(3 a)}^{2} - 12 a b + {(2 b)}^{2}}{6}}{\frac{1}{4} a^{2} - \frac{1}{9} b^{2}} + \frac{6 b}{\frac{3}{4} a + \frac{1}{2} b}$

Step 1.1.10.6.3

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$12 a b = 2 \cdot (3 a) \cdot (2 b)$

Step 1.1.10.6.4

Rewrite the polynomial.

$\frac{\frac{{(3 a)}^{2} - 2 \cdot (3 a) \cdot (2 b) + {(2 b)}^{2}}{6}}{\frac{1}{4} a^{2} - \frac{1}{9} b^{2}} + \frac{6 b}{\frac{3}{4} a + \frac{1}{2} b}$

Step 1.1.10.6.5

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = 3 a$ and $b = 2 b$ .

$\frac{\frac{{(3 a - 2 b)}^{2}}{6}}{\frac{1}{4} a^{2} - \frac{1}{9} b^{2}} + \frac{6 b}{\frac{3}{4} a + \frac{1}{2} b}$

Step 1.2

Simplify the denominator.

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Step 1.2.1

Rewrite $\frac{1}{4} a^{2}$ as ${(\frac{1}{2} a)}^{2}$ .

$\frac{\frac{{(3 a - 2 b)}^{2}}{6}}{{(\frac{1}{2} a)}^{2} - \frac{1}{9} b^{2}} + \frac{6 b}{\frac{3}{4} a + \frac{1}{2} b}$

Step 1.2.2

Rewrite $\frac{1}{9} b^{2}$ as ${(\frac{1}{3} b)}^{2}$ .

$\frac{\frac{{(3 a - 2 b)}^{2}}{6}}{{(\frac{1}{2} a)}^{2} - {(\frac{1}{3} b)}^{2}} + \frac{6 b}{\frac{3}{4} a + \frac{1}{2} b}$

Step 1.2.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = \frac{1}{2} a$ and $b = \frac{1}{3} b$ .

$\frac{\frac{{(3 a - 2 b)}^{2}}{6}}{(\frac{1}{2} a + \frac{1}{3} b) (\frac{1}{2} a - (\frac{1}{3} b))} + \frac{6 b}{\frac{3}{4} a + \frac{1}{2} b}$

Step 1.2.4

Simplify.

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Step 1.2.4.1

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

$\frac{\frac{{(3 a - 2 b)}^{2}}{6}}{(\frac{a}{2} + \frac{1}{3} b) (\frac{1}{2} a - (\frac{1}{3} b))} + \frac{6 b}{\frac{3}{4} a + \frac{1}{2} b}$

Step 1.2.4.2

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

$\frac{\frac{{(3 a - 2 b)}^{2}}{6}}{(\frac{a}{2} + \frac{b}{3}) (\frac{1}{2} a - (\frac{1}{3} b))} + \frac{6 b}{\frac{3}{4} a + \frac{1}{2} b}$

Step 1.2.4.3

To write $\frac{a}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{\frac{{(3 a - 2 b)}^{2}}{6}}{(\frac{a}{2} \cdot \frac{3}{3} + \frac{b}{3}) (\frac{1}{2} a - (\frac{1}{3} b))} + \frac{6 b}{\frac{3}{4} a + \frac{1}{2} b}$

Step 1.2.4.4

To write $\frac{b}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{\frac{{(3 a - 2 b)}^{2}}{6}}{(\frac{a}{2} \cdot \frac{3}{3} + \frac{b}{3} \cdot \frac{2}{2}) (\frac{1}{2} a - (\frac{1}{3} b))} + \frac{6 b}{\frac{3}{4} a + \frac{1}{2} b}$

Step 1.2.4.5

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.2.4.5.1

Multiply $\frac{a}{2}$ by $\frac{3}{3}$ .

$\frac{\frac{{(3 a - 2 b)}^{2}}{6}}{(\frac{a \cdot 3}{2 \cdot 3} + \frac{b}{3} \cdot \frac{2}{2}) (\frac{1}{2} a - (\frac{1}{3} b))} + \frac{6 b}{\frac{3}{4} a + \frac{1}{2} b}$

Step 1.2.4.5.2

Multiply $2$ by $3$ .

$\frac{\frac{{(3 a - 2 b)}^{2}}{6}}{(\frac{a \cdot 3}{6} + \frac{b}{3} \cdot \frac{2}{2}) (\frac{1}{2} a - (\frac{1}{3} b))} + \frac{6 b}{\frac{3}{4} a + \frac{1}{2} b}$

Step 1.2.4.5.3

Multiply $\frac{b}{3}$ by $\frac{2}{2}$ .

$\frac{\frac{{(3 a - 2 b)}^{2}}{6}}{(\frac{a \cdot 3}{6} + \frac{b \cdot 2}{3 \cdot 2}) (\frac{1}{2} a - (\frac{1}{3} b))} + \frac{6 b}{\frac{3}{4} a + \frac{1}{2} b}$

Step 1.2.4.5.4

Multiply $3$ by $2$ .

$\frac{\frac{{(3 a - 2 b)}^{2}}{6}}{(\frac{a \cdot 3}{6} + \frac{b \cdot 2}{6}) (\frac{1}{2} a - (\frac{1}{3} b))} + \frac{6 b}{\frac{3}{4} a + \frac{1}{2} b}$

Step 1.2.4.6

Combine the numerators over the common denominator.

$\frac{\frac{{(3 a - 2 b)}^{2}}{6}}{\frac{a \cdot 3 + b \cdot 2}{6} (\frac{1}{2} a - (\frac{1}{3} b))} + \frac{6 b}{\frac{3}{4} a + \frac{1}{2} b}$

Step 1.2.4.7

Rewrite $\frac{a \cdot 3 + b \cdot 2}{6}$ in a factored form.

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Step 1.2.4.7.1

Move $3$ to the left of $a$ .

$\frac{\frac{{(3 a - 2 b)}^{2}}{6}}{\frac{3 \cdot a + b \cdot 2}{6} (\frac{1}{2} a - (\frac{1}{3} b))} + \frac{6 b}{\frac{3}{4} a + \frac{1}{2} b}$

Step 1.2.4.7.2

Move $2$ to the left of $b$ .

$\frac{\frac{{(3 a - 2 b)}^{2}}{6}}{\frac{3 a + 2 b}{6} (\frac{1}{2} a - (\frac{1}{3} b))} + \frac{6 b}{\frac{3}{4} a + \frac{1}{2} b}$

Step 1.2.4.8

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

$\frac{\frac{{(3 a - 2 b)}^{2}}{6}}{\frac{3 a + 2 b}{6} (\frac{a}{2} - (\frac{1}{3} b))} + \frac{6 b}{\frac{3}{4} a + \frac{1}{2} b}$

Step 1.2.4.9

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

$\frac{\frac{{(3 a - 2 b)}^{2}}{6}}{\frac{3 a + 2 b}{6} (\frac{a}{2} - \frac{b}{3})} + \frac{6 b}{\frac{3}{4} a + \frac{1}{2} b}$

Step 1.2.4.10

To write $\frac{a}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{\frac{{(3 a - 2 b)}^{2}}{6}}{\frac{3 a + 2 b}{6} (\frac{a}{2} \cdot \frac{3}{3} - \frac{b}{3})} + \frac{6 b}{\frac{3}{4} a + \frac{1}{2} b}$

Step 1.2.4.11

To write $- \frac{b}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{\frac{{(3 a - 2 b)}^{2}}{6}}{\frac{3 a + 2 b}{6} (\frac{a}{2} \cdot \frac{3}{3} - \frac{b}{3} \cdot \frac{2}{2})} + \frac{6 b}{\frac{3}{4} a + \frac{1}{2} b}$

Step 1.2.4.12

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.2.4.12.1

Multiply $\frac{a}{2}$ by $\frac{3}{3}$ .

$\frac{\frac{{(3 a - 2 b)}^{2}}{6}}{\frac{3 a + 2 b}{6} (\frac{a \cdot 3}{2 \cdot 3} - \frac{b}{3} \cdot \frac{2}{2})} + \frac{6 b}{\frac{3}{4} a + \frac{1}{2} b}$

Step 1.2.4.12.2

Multiply $2$ by $3$ .

$\frac{\frac{{(3 a - 2 b)}^{2}}{6}}{\frac{3 a + 2 b}{6} (\frac{a \cdot 3}{6} - \frac{b}{3} \cdot \frac{2}{2})} + \frac{6 b}{\frac{3}{4} a + \frac{1}{2} b}$

Step 1.2.4.12.3

Multiply $\frac{b}{3}$ by $\frac{2}{2}$ .

$\frac{\frac{{(3 a - 2 b)}^{2}}{6}}{\frac{3 a + 2 b}{6} (\frac{a \cdot 3}{6} - \frac{b \cdot 2}{3 \cdot 2})} + \frac{6 b}{\frac{3}{4} a + \frac{1}{2} b}$

Step 1.2.4.12.4

Multiply $3$ by $2$ .

$\frac{\frac{{(3 a - 2 b)}^{2}}{6}}{\frac{3 a + 2 b}{6} (\frac{a \cdot 3}{6} - \frac{b \cdot 2}{6})} + \frac{6 b}{\frac{3}{4} a + \frac{1}{2} b}$

Step 1.2.4.13

Combine the numerators over the common denominator.

$\frac{\frac{{(3 a - 2 b)}^{2}}{6}}{\frac{3 a + 2 b}{6} \cdot \frac{a \cdot 3 - b \cdot 2}{6}} + \frac{6 b}{\frac{3}{4} a + \frac{1}{2} b}$

Step 1.2.4.14

Rewrite $\frac{a \cdot 3 - b \cdot 2}{6}$ in a factored form.

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Step 1.2.4.14.1

Move $3$ to the left of $a$ .

$\frac{\frac{{(3 a - 2 b)}^{2}}{6}}{\frac{3 a + 2 b}{6} \cdot \frac{3 \cdot a - b \cdot 2}{6}} + \frac{6 b}{\frac{3}{4} a + \frac{1}{2} b}$

Step 1.2.4.14.2

Multiply $2$ by $- 1$ .

$\frac{\frac{{(3 a - 2 b)}^{2}}{6}}{\frac{3 a + 2 b}{6} \cdot \frac{3 a - 2 b}{6}} + \frac{6 b}{\frac{3}{4} a + \frac{1}{2} b}$

Step 1.3

Multiply $\frac{3 a + 2 b}{6}$ by $\frac{3 a - 2 b}{6}$ .

$\frac{\frac{{(3 a - 2 b)}^{2}}{6}}{\frac{(3 a + 2 b) (3 a - 2 b)}{6 \cdot 6}} + \frac{6 b}{\frac{3}{4} a + \frac{1}{2} b}$

Step 1.4

Multiply $6$ by $6$ .

$\frac{\frac{{(3 a - 2 b)}^{2}}{6}}{\frac{(3 a + 2 b) (3 a - 2 b)}{36}} + \frac{6 b}{\frac{3}{4} a + \frac{1}{2} b}$

Step 1.5

Multiply the numerator by the reciprocal of the denominator.

$\frac{{(3 a - 2 b)}^{2}}{6} \cdot \frac{36}{(3 a + 2 b) (3 a - 2 b)} + \frac{6 b}{\frac{3}{4} a + \frac{1}{2} b}$

Step 1.6

Combine.

$\frac{{(3 a - 2 b)}^{2} \cdot 36}{6 ((3 a + 2 b) (3 a - 2 b))} + \frac{6 b}{\frac{3}{4} a + \frac{1}{2} b}$

Step 1.7

Cancel the common factor of ${(3 a - 2 b)}^{2}$ and $3 a - 2 b$ .

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Step 1.7.1

Factor $3 a - 2 b$ out of ${(3 a - 2 b)}^{2} \cdot 36$ .

$\frac{(3 a - 2 b) ((3 a - 2 b) \cdot 36)}{6 ((3 a + 2 b) (3 a - 2 b))} + \frac{6 b}{\frac{3}{4} a + \frac{1}{2} b}$

Step 1.7.2

Cancel the common factors.

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Step 1.7.2.1

Factor $3 a - 2 b$ out of $6 ((3 a + 2 b) (3 a - 2 b))$ .

$\frac{(3 a - 2 b) ((3 a - 2 b) \cdot 36)}{(3 a - 2 b) (6 (3 a + 2 b))} + \frac{6 b}{\frac{3}{4} a + \frac{1}{2} b}$

Step 1.7.2.2

Cancel the common factor.

$\frac{(3 a - 2 b) ((3 a - 2 b) \cdot 36)}{(3 a - 2 b) (6 (3 a + 2 b))} + \frac{6 b}{\frac{3}{4} a + \frac{1}{2} b}$

Step 1.7.2.3

Rewrite the expression.

$\frac{(3 a - 2 b) \cdot 36}{6 (3 a + 2 b)} + \frac{6 b}{\frac{3}{4} a + \frac{1}{2} b}$

Step 1.8

Cancel the common factor of $36$ and $6$ .

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Step 1.8.1

Factor $6$ out of $(3 a - 2 b) \cdot 36$ .

$\frac{6 ((3 a - 2 b) \cdot 6)}{6 (3 a + 2 b)} + \frac{6 b}{\frac{3}{4} a + \frac{1}{2} b}$

Step 1.8.2

Cancel the common factors.

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Step 1.8.2.1

Cancel the common factor.

$\frac{6 ((3 a - 2 b) \cdot 6)}{6 (3 a + 2 b)} + \frac{6 b}{\frac{3}{4} a + \frac{1}{2} b}$

Step 1.8.2.2

Rewrite the expression.

$\frac{(3 a - 2 b) \cdot 6}{3 a + 2 b} + \frac{6 b}{\frac{3}{4} a + \frac{1}{2} b}$

Step 1.9

Move $6$ to the left of $3 a - 2 b$ .

$\frac{6 \cdot (3 a - 2 b)}{3 a + 2 b} + \frac{6 b}{\frac{3}{4} a + \frac{1}{2} b}$

Step 1.10

Simplify the denominator.

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Step 1.10.1

Combine $\frac{3}{4}$ and $a$ .

$\frac{6 (3 a - 2 b)}{3 a + 2 b} + \frac{6 b}{\frac{3 a}{4} + \frac{1}{2} b}$

Step 1.10.2

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

$\frac{6 (3 a - 2 b)}{3 a + 2 b} + \frac{6 b}{\frac{3 a}{4} + \frac{b}{2}}$

Step 1.10.3

To write $\frac{b}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{6 (3 a - 2 b)}{3 a + 2 b} + \frac{6 b}{\frac{3 a}{4} + \frac{b}{2} \cdot \frac{2}{2}}$

Step 1.10.4

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.10.4.1

Multiply $\frac{b}{2}$ by $\frac{2}{2}$ .

$\frac{6 (3 a - 2 b)}{3 a + 2 b} + \frac{6 b}{\frac{3 a}{4} + \frac{b \cdot 2}{2 \cdot 2}}$

Step 1.10.4.2

Multiply $2$ by $2$ .

$\frac{6 (3 a - 2 b)}{3 a + 2 b} + \frac{6 b}{\frac{3 a}{4} + \frac{b \cdot 2}{4}}$

Step 1.10.5

Combine the numerators over the common denominator.

$\frac{6 (3 a - 2 b)}{3 a + 2 b} + \frac{6 b}{\frac{3 a + b \cdot 2}{4}}$

Step 1.10.6

Move $2$ to the left of $b$ .

$\frac{6 (3 a - 2 b)}{3 a + 2 b} + \frac{6 b}{\frac{3 a + 2 b}{4}}$

Step 1.11

Multiply the numerator by the reciprocal of the denominator.

$\frac{6 (3 a - 2 b)}{3 a + 2 b} + 6 b \frac{4}{3 a + 2 b}$

Step 1.12

Multiply $6 b \frac{4}{3 a + 2 b}$ .

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Step 1.12.1

Combine $6$ and $\frac{4}{3 a + 2 b}$ .

$\frac{6 (3 a - 2 b)}{3 a + 2 b} + b \frac{6 \cdot 4}{3 a + 2 b}$

Step 1.12.2

Multiply $6$ by $4$ .

$\frac{6 (3 a - 2 b)}{3 a + 2 b} + b \frac{24}{3 a + 2 b}$

Step 1.12.3

Combine $b$ and $\frac{24}{3 a + 2 b}$ .

$\frac{6 (3 a - 2 b)}{3 a + 2 b} + \frac{b \cdot 24}{3 a + 2 b}$

Step 1.13

Move $24$ to the left of $b$ .

$\frac{6 (3 a - 2 b)}{3 a + 2 b} + \frac{24 b}{3 a + 2 b}$

Step 2

Combine the numerators over the common denominator.

$\frac{6 (3 a - 2 b) + 24 b}{3 a + 2 b}$

Step 3

Simplify the numerator.

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Step 3.1

Simplify each term.

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Step 3.1.1

Apply the distributive property.

$\frac{6 (3 a) + 6 (- 2 b) + 24 b}{3 a + 2 b}$

Step 3.1.2

Multiply $3$ by $6$ .

$\frac{18 a + 6 (- 2 b) + 24 b}{3 a + 2 b}$

Step 3.1.3

Multiply $- 2$ by $6$ .

$\frac{18 a - 12 b + 24 b}{3 a + 2 b}$

Step 3.2

Add $- 12 b$ and $24 b$ .

$\frac{18 a + 12 b}{3 a + 2 b}$

Step 4

Factor $6$ out of $18 a + 12 b$ .

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Step 4.1

Factor $6$ out of $18 a$ .

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

Step 4.2

Factor $6$ out of $12 b$ .

$\frac{6 (3 a) + 6 (2 b)}{3 a + 2 b}$

Step 4.3

Factor $6$ out of $6 (3 a) + 6 (2 b)$ .

$\frac{6 (3 a + 2 b)}{3 a + 2 b}$

Step 5

Cancel the common factor of $3 a + 2 b$ .

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Step 5.1

Cancel the common factor.

$\frac{6 (3 a + 2 b)}{3 a + 2 b}$

Step 5.2

Divide $6$ by $1$ .

$6$