Algebra Examples

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

Step 1

Simplify terms.

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Step 1.1

Simplify each term.

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Step 1.1.1

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

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

Step 1.1.2

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

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

Step 1.1.3

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

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Step 1.1.3.1

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

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

Step 1.1.3.2

Multiply $3$ by $2$ .

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

Step 1.1.3.3

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

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

Step 1.1.3.4

Multiply $2$ by $3$ .

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

Step 1.1.3.5

Reorder the factors of $6 b a$ .

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

Step 1.1.4

Combine the numerators over the common denominator.

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

Step 1.1.5

Simplify the numerator.

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Step 1.1.5.1

Apply the distributive property.

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

Step 1.1.5.2

Multiply $3$ by $2$ .

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

Step 1.1.5.3

Multiply $3$ by $3$ .

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

Step 1.1.5.4

Apply the distributive property.

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

Step 1.1.5.5

Multiply $b$ by $b$ by adding the exponents.

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Step 1.1.5.5.1

Move $b$ .

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

Step 1.1.5.5.2

Multiply $b$ by $b$ .

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

Step 1.1.5.6

Apply the distributive property.

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

Step 1.1.5.7

Multiply $2$ by $2$ .

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

Step 1.1.5.8

Multiply $2$ by $- 3$ .

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

Step 1.1.5.9

Apply the distributive property.

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

Step 1.1.5.10

Multiply $a$ by $a$ by adding the exponents.

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Step 1.1.5.10.1

Move $a$ .

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

Step 1.1.5.10.2

Multiply $a$ by $a$ .

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

Step 1.1.5.11

Subtract $6 b a$ from $6 a b$ .

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Step 1.1.5.11.1

Move $b$ .

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

Step 1.1.5.11.2

Subtract $6 a b$ from $6 a b$ .

${(\frac{9 b^{2} + 4 a^{2} + 0}{6 a b})}^{2} - {(\frac{2 a + 3 b}{2 a} - \frac{2 a - 3 b}{3 b})}^{2}$

Step 1.1.5.12

Add $9 b^{2} + 4 a^{2}$ and $0$ .

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

Step 1.1.6

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 1.1.6.1

Apply the product rule to $\frac{9 b^{2} + 4 a^{2}}{6 a b}$ .

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

Step 1.1.6.2

Apply the product rule to $6 a b$ .

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

Step 1.1.6.3

Apply the product rule to $6 a$ .

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

Step 1.1.7

Raise $6$ to the power of $2$ .

$\frac{{(9 b^{2} + 4 a^{2})}^{2}}{36 a^{2} b^{2}} - {(\frac{2 a + 3 b}{2 a} - \frac{2 a - 3 b}{3 b})}^{2}$

Step 1.1.8

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

$\frac{{(9 b^{2} + 4 a^{2})}^{2}}{36 a^{2} b^{2}} - {(\frac{2 a + 3 b}{2 a} \cdot \frac{3 b}{3 b} - \frac{2 a - 3 b}{3 b})}^{2}$

Step 1.1.9

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

$\frac{{(9 b^{2} + 4 a^{2})}^{2}}{36 a^{2} b^{2}} - {(\frac{2 a + 3 b}{2 a} \cdot \frac{3 b}{3 b} - \frac{2 a - 3 b}{3 b} \cdot \frac{2 a}{2 a})}^{2}$

Step 1.1.10

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

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Step 1.1.10.1

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

$\frac{{(9 b^{2} + 4 a^{2})}^{2}}{36 a^{2} b^{2}} - {(\frac{(2 a + 3 b) (3 b)}{2 a (3 b)} - \frac{2 a - 3 b}{3 b} \cdot \frac{2 a}{2 a})}^{2}$

Step 1.1.10.2

Multiply $3$ by $2$ .

$\frac{{(9 b^{2} + 4 a^{2})}^{2}}{36 a^{2} b^{2}} - {(\frac{(2 a + 3 b) (3 b)}{6 a b} - \frac{2 a - 3 b}{3 b} \cdot \frac{2 a}{2 a})}^{2}$

Step 1.1.10.3

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

$\frac{{(9 b^{2} + 4 a^{2})}^{2}}{36 a^{2} b^{2}} - {(\frac{(2 a + 3 b) (3 b)}{6 a b} - \frac{(2 a - 3 b) (2 a)}{3 b (2 a)})}^{2}$

Step 1.1.10.4

Multiply $2$ by $3$ .

$\frac{{(9 b^{2} + 4 a^{2})}^{2}}{36 a^{2} b^{2}} - {(\frac{(2 a + 3 b) (3 b)}{6 a b} - \frac{(2 a - 3 b) (2 a)}{6 b a})}^{2}$

Step 1.1.10.5

Reorder the factors of $6 b a$ .

$\frac{{(9 b^{2} + 4 a^{2})}^{2}}{36 a^{2} b^{2}} - {(\frac{(2 a + 3 b) (3 b)}{6 a b} - \frac{(2 a - 3 b) (2 a)}{6 a b})}^{2}$

Step 1.1.11

Combine the numerators over the common denominator.

$\frac{{(9 b^{2} + 4 a^{2})}^{2}}{36 a^{2} b^{2}} - {(\frac{(2 a + 3 b) (3 b) - (2 a - 3 b) (2 a)}{6 a b})}^{2}$

Step 1.1.12

Simplify the numerator.

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Step 1.1.12.1

Apply the distributive property.

$\frac{{(9 b^{2} + 4 a^{2})}^{2}}{36 a^{2} b^{2}} - {(\frac{(2 a \cdot 3 + 3 b \cdot 3) b - (2 a - 3 b) (2 a)}{6 a b})}^{2}$

Step 1.1.12.2

Multiply $3$ by $2$ .

$\frac{{(9 b^{2} + 4 a^{2})}^{2}}{36 a^{2} b^{2}} - {(\frac{(6 a + 3 b \cdot 3) b - (2 a - 3 b) (2 a)}{6 a b})}^{2}$

Step 1.1.12.3

Multiply $3$ by $3$ .

$\frac{{(9 b^{2} + 4 a^{2})}^{2}}{36 a^{2} b^{2}} - {(\frac{(6 a + 9 b) b - (2 a - 3 b) (2 a)}{6 a b})}^{2}$

Step 1.1.12.4

Apply the distributive property.

$\frac{{(9 b^{2} + 4 a^{2})}^{2}}{36 a^{2} b^{2}} - {(\frac{6 a b + 9 b \cdot b - (2 a - 3 b) (2 a)}{6 a b})}^{2}$

Step 1.1.12.5

Multiply $b$ by $b$ by adding the exponents.

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Step 1.1.12.5.1

Move $b$ .

$\frac{{(9 b^{2} + 4 a^{2})}^{2}}{36 a^{2} b^{2}} - {(\frac{6 a b + 9 (b \cdot b) - (2 a - 3 b) (2 a)}{6 a b})}^{2}$

Step 1.1.12.5.2

Multiply $b$ by $b$ .

$\frac{{(9 b^{2} + 4 a^{2})}^{2}}{36 a^{2} b^{2}} - {(\frac{6 a b + 9 b^{2} - (2 a - 3 b) (2 a)}{6 a b})}^{2}$

Step 1.1.12.6

Apply the distributive property.

$\frac{{(9 b^{2} + 4 a^{2})}^{2}}{36 a^{2} b^{2}} - {(\frac{6 a b + 9 b^{2} + (- (2 a) - (- 3 b)) (2 a)}{6 a b})}^{2}$

Step 1.1.12.7

Multiply $2$ by $- 1$ .

$\frac{{(9 b^{2} + 4 a^{2})}^{2}}{36 a^{2} b^{2}} - {(\frac{6 a b + 9 b^{2} + (- 2 a - (- 3 b)) (2 a)}{6 a b})}^{2}$

Step 1.1.12.8

Multiply $- 3$ by $- 1$ .

$\frac{{(9 b^{2} + 4 a^{2})}^{2}}{36 a^{2} b^{2}} - {(\frac{6 a b + 9 b^{2} + (- 2 a + 3 b) (2 a)}{6 a b})}^{2}$

Step 1.1.12.9

Apply the distributive property.

$\frac{{(9 b^{2} + 4 a^{2})}^{2}}{36 a^{2} b^{2}} - {(\frac{6 a b + 9 b^{2} - 2 a (2 a) + 3 b (2 a)}{6 a b})}^{2}$

Step 1.1.12.10

Rewrite using the commutative property of multiplication.

$\frac{{(9 b^{2} + 4 a^{2})}^{2}}{36 a^{2} b^{2}} - {(\frac{6 a b + 9 b^{2} - 2 \cdot 2 a \cdot a + 3 b (2 a)}{6 a b})}^{2}$

Step 1.1.12.11

Rewrite using the commutative property of multiplication.

$\frac{{(9 b^{2} + 4 a^{2})}^{2}}{36 a^{2} b^{2}} - {(\frac{6 a b + 9 b^{2} - 2 \cdot 2 a \cdot a + 3 \cdot 2 b a}{6 a b})}^{2}$

Step 1.1.12.12

Simplify each term.

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Step 1.1.12.12.1

Multiply $a$ by $a$ by adding the exponents.

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Step 1.1.12.12.1.1

Move $a$ .

$\frac{{(9 b^{2} + 4 a^{2})}^{2}}{36 a^{2} b^{2}} - {(\frac{6 a b + 9 b^{2} - 2 \cdot 2 (a \cdot a) + 3 \cdot 2 b a}{6 a b})}^{2}$

Step 1.1.12.12.1.2

Multiply $a$ by $a$ .

$\frac{{(9 b^{2} + 4 a^{2})}^{2}}{36 a^{2} b^{2}} - {(\frac{6 a b + 9 b^{2} - 2 \cdot 2 a^{2} + 3 \cdot 2 b a}{6 a b})}^{2}$

Step 1.1.12.12.2

Multiply $- 2$ by $2$ .

$\frac{{(9 b^{2} + 4 a^{2})}^{2}}{36 a^{2} b^{2}} - {(\frac{6 a b + 9 b^{2} - 4 a^{2} + 3 \cdot 2 b a}{6 a b})}^{2}$

Step 1.1.12.12.3

Multiply $3$ by $2$ .

$\frac{{(9 b^{2} + 4 a^{2})}^{2}}{36 a^{2} b^{2}} - {(\frac{6 a b + 9 b^{2} - 4 a^{2} + 6 b a}{6 a b})}^{2}$

Step 1.1.12.13

Add $6 a b$ and $6 b a$ .

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Step 1.1.12.13.1

Move $b$ .

$\frac{{(9 b^{2} + 4 a^{2})}^{2}}{36 a^{2} b^{2}} - {(\frac{9 b^{2} - 4 a^{2} + 6 a b + 6 a b}{6 a b})}^{2}$

Step 1.1.12.13.2

Add $6 a b$ and $6 a b$ .

$\frac{{(9 b^{2} + 4 a^{2})}^{2}}{36 a^{2} b^{2}} - {(\frac{9 b^{2} - 4 a^{2} + 12 a b}{6 a b})}^{2}$

Step 1.1.13

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 1.1.13.1

Apply the product rule to $\frac{9 b^{2} - 4 a^{2} + 12 a b}{6 a b}$ .

$\frac{{(9 b^{2} + 4 a^{2})}^{2}}{36 a^{2} b^{2}} - \frac{{(9 b^{2} - 4 a^{2} + 12 a b)}^{2}}{{(6 a b)}^{2}}$

Step 1.1.13.2

Apply the product rule to $6 a b$ .

$\frac{{(9 b^{2} + 4 a^{2})}^{2}}{36 a^{2} b^{2}} - \frac{{(9 b^{2} - 4 a^{2} + 12 a b)}^{2}}{{(6 a)}^{2} b^{2}}$

Step 1.1.13.3

Apply the product rule to $6 a$ .

$\frac{{(9 b^{2} + 4 a^{2})}^{2}}{36 a^{2} b^{2}} - \frac{{(9 b^{2} - 4 a^{2} + 12 a b)}^{2}}{6^{2} a^{2} b^{2}}$

Step 1.1.14

Raise $6$ to the power of $2$ .

$\frac{{(9 b^{2} + 4 a^{2})}^{2}}{36 a^{2} b^{2}} - \frac{{(9 b^{2} - 4 a^{2} + 12 a b)}^{2}}{36 a^{2} b^{2}}$

Step 1.2

Combine the numerators over the common denominator.

$\frac{{(9 b^{2} + 4 a^{2})}^{2} - {(9 b^{2} - 4 a^{2} + 12 a b)}^{2}}{36 a^{2} b^{2}}$

Step 2

Simplify the numerator.

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Step 2.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 9 b^{2} + 4 a^{2}$ and $b = 9 b^{2} - 4 a^{2} + 12 a b$ .

$\frac{(9 b^{2} + 4 a^{2} + 9 b^{2} - 4 a^{2} + 12 a b) (9 b^{2} + 4 a^{2} - (9 b^{2} - 4 a^{2} + 12 a b))}{36 a^{2} b^{2}}$

Step 2.2

Simplify.

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Step 2.2.1

Add $9 b^{2}$ and $9 b^{2}$ .

$\frac{(4 a^{2} + 18 b^{2} - 4 a^{2} + 12 a b) (9 b^{2} + 4 a^{2} - (9 b^{2} - 4 a^{2} + 12 a b))}{36 a^{2} b^{2}}$

Step 2.2.2

Subtract $4 a^{2}$ from $4 a^{2}$ .

$\frac{(18 b^{2} + 0 + 12 a b) (9 b^{2} + 4 a^{2} - (9 b^{2} - 4 a^{2} + 12 a b))}{36 a^{2} b^{2}}$

Step 2.2.3

Add $18 b^{2}$ and $0$ .

$\frac{(18 b^{2} + 12 a b) (9 b^{2} + 4 a^{2} - (9 b^{2} - 4 a^{2} + 12 a b))}{36 a^{2} b^{2}}$

Step 2.2.4

Factor $6 b$ out of $18 b^{2} + 12 a b$ .

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Step 2.2.4.1

Factor $6 b$ out of $18 b^{2}$ .

$\frac{(6 b (3 b) + 12 a b) (9 b^{2} + 4 a^{2} - (9 b^{2} - 4 a^{2} + 12 a b))}{36 a^{2} b^{2}}$

Step 2.2.4.2

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

$\frac{(6 b (3 b) + 6 b (2 a)) (9 b^{2} + 4 a^{2} - (9 b^{2} - 4 a^{2} + 12 a b))}{36 a^{2} b^{2}}$

Step 2.2.4.3

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

$\frac{6 b (3 b + 2 a) (9 b^{2} + 4 a^{2} - (9 b^{2} - 4 a^{2} + 12 a b))}{36 a^{2} b^{2}}$

Step 2.2.5

Apply the distributive property.

$\frac{6 b (3 b + 2 a) (9 b^{2} + 4 a^{2} - (9 b^{2}) - (- 4 a^{2}) - (12 a b))}{36 a^{2} b^{2}}$

Step 2.2.6

Simplify.

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Step 2.2.6.1

Multiply $9$ by $- 1$ .

$\frac{6 b (3 b + 2 a) (9 b^{2} + 4 a^{2} - 9 b^{2} - (- 4 a^{2}) - (12 a b))}{36 a^{2} b^{2}}$

Step 2.2.6.2

Multiply $- 4$ by $- 1$ .

$\frac{6 b (3 b + 2 a) (9 b^{2} + 4 a^{2} - 9 b^{2} + 4 a^{2} - (12 a b))}{36 a^{2} b^{2}}$

Step 2.2.6.3

Multiply $12$ by $- 1$ .

$\frac{6 b (3 b + 2 a) (9 b^{2} + 4 a^{2} - 9 b^{2} + 4 a^{2} - 12 (a b))}{36 a^{2} b^{2}}$

Step 2.2.7

Subtract $9 b^{2}$ from $9 b^{2}$ .

$\frac{6 b (3 b + 2 a) (4 a^{2} + 0 + 4 a^{2} - 12 a b)}{36 a^{2} b^{2}}$

Step 2.2.8

Add $4 a^{2}$ and $0$ .

$\frac{6 b (3 b + 2 a) (4 a^{2} + 4 a^{2} - 12 a b)}{36 a^{2} b^{2}}$

Step 2.2.9

Add $4 a^{2}$ and $4 a^{2}$ .

$\frac{6 b (3 b + 2 a) (8 a^{2} - 12 a b)}{36 a^{2} b^{2}}$

Step 2.2.10

Factor $4 a$ out of $8 a^{2} - 12 a b$ .

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Step 2.2.10.1

Factor $4 a$ out of $8 a^{2}$ .

$\frac{6 b (3 b + 2 a) (4 a (2 a) - 12 a b)}{36 a^{2} b^{2}}$

Step 2.2.10.2

Factor $4 a$ out of $- 12 a b$ .

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

Step 2.2.10.3

Factor $4 a$ out of $4 a (2 a) + 4 a (- 3 b)$ .

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

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

Step 2.2.11

Multiply $4$ by $6$ .

$\frac{24 b (3 b + 2 a) a (2 a - 3 b)}{36 a^{2} b^{2}}$

Step 3

Reduce the expression by cancelling the common factors.

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Step 3.1

Cancel the common factor of $24$ and $36$ .

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Step 3.1.1

Factor $12$ out of $24 b (3 b + 2 a) a (2 a - 3 b)$ .

$\frac{12 (2 b (3 b + 2 a) a (2 a - 3 b))}{36 a^{2} b^{2}}$

Step 3.1.2

Cancel the common factors.

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Step 3.1.2.1

Factor $12$ out of $36 a^{2} b^{2}$ .

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

Step 3.1.2.2

Cancel the common factor.

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

Step 3.1.2.3

Rewrite the expression.

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

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

Cancel the common factors.

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Step 3.2.2.1

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

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

Step 3.2.2.2

Cancel the common factor.

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

Step 3.2.2.3

Rewrite the expression.

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

Step 3.3

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

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Step 3.3.1

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

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

Step 3.3.2

Cancel the common factors.

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Step 3.3.2.1

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

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

Step 3.3.2.2

Cancel the common factor.

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

Step 3.3.2.3

Rewrite the expression.

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

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