Basic Math Examples

$\frac{5 a}{a - b} + \frac{a b}{a^{2} - b^{2}} + \frac{4 b}{a + b}$

Step 1

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

$\frac{5 a}{a - b} + \frac{a b}{(a + b) (a - b)} + \frac{4 b}{a + b}$

Step 2

To write $\frac{5 a}{a - b}$ as a fraction with a common denominator, multiply by $\frac{a + b}{a + b}$ .

$\frac{5 a}{a - b} \cdot \frac{a + b}{a + b} + \frac{a b}{(a + b) (a - b)} + \frac{4 b}{a + b}$

Step 3

Write each expression with a common denominator of $(a - b) (a + b)$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{5 a}{a - b}$ by $\frac{a + b}{a + b}$ .

$\frac{5 a (a + b)}{(a - b) (a + b)} + \frac{a b}{(a + b) (a - b)} + \frac{4 b}{a + b}$

Step 3.2

Reorder the factors of $(a - b) (a + b)$ .

$\frac{5 a (a + b)}{(a + b) (a - b)} + \frac{a b}{(a + b) (a - b)} + \frac{4 b}{a + b}$

Step 4

Combine the numerators over the common denominator.

$\frac{5 a (a + b) + a b}{(a + b) (a - b)} + \frac{4 b}{a + b}$

Step 5

Simplify the numerator.

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Step 5.1

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

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Step 5.1.1

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

$\frac{a (5 (a + b)) + a b}{(a + b) (a - b)} + \frac{4 b}{a + b}$

Step 5.1.2

Factor $a$ out of $a b$ .

$\frac{a (5 (a + b)) + a (b)}{(a + b) (a - b)} + \frac{4 b}{a + b}$

Step 5.1.3

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

$\frac{a (5 (a + b) + b)}{(a + b) (a - b)} + \frac{4 b}{a + b}$

Step 5.2

Apply the distributive property.

$\frac{a (5 a + 5 b + b)}{(a + b) (a - b)} + \frac{4 b}{a + b}$

Step 5.3

Add $5 b$ and $b$ .

$\frac{a (5 a + 6 b)}{(a + b) (a - b)} + \frac{4 b}{a + b}$

Step 6

To write $\frac{4 b}{a + b}$ as a fraction with a common denominator, multiply by $\frac{a - b}{a - b}$ .

$\frac{a (5 a + 6 b)}{(a + b) (a - b)} + \frac{4 b}{a + b} \cdot \frac{a - b}{a - b}$

Step 7

Simplify terms.

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Step 7.1

Multiply $\frac{4 b}{a + b}$ by $\frac{a - b}{a - b}$ .

$\frac{a (5 a + 6 b)}{(a + b) (a - b)} + \frac{4 b (a - b)}{(a + b) (a - b)}$

Step 7.2

Combine the numerators over the common denominator.

$\frac{a (5 a + 6 b) + 4 b (a - b)}{(a + b) (a - b)}$

Step 8

Simplify the numerator.

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Step 8.1

Apply the distributive property.

$\frac{a (5 a) + a (6 b) + 4 b (a - b)}{(a + b) (a - b)}$

Step 8.2

Rewrite using the commutative property of multiplication.

$\frac{5 a \cdot a + a (6 b) + 4 b (a - b)}{(a + b) (a - b)}$

Step 8.3

Rewrite using the commutative property of multiplication.

$\frac{5 a \cdot a + 6 a b + 4 b (a - b)}{(a + b) (a - b)}$

Step 8.4

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

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Step 8.4.1

Move $a$ .

$\frac{5 (a \cdot a) + 6 a b + 4 b (a - b)}{(a + b) (a - b)}$

Step 8.4.2

Multiply $a$ by $a$ .

$\frac{5 a^{2} + 6 a b + 4 b (a - b)}{(a + b) (a - b)}$

Step 8.5

Apply the distributive property.

$\frac{5 a^{2} + 6 a b + 4 b a + 4 b (- b)}{(a + b) (a - b)}$

Step 8.6

Rewrite using the commutative property of multiplication.

$\frac{5 a^{2} + 6 a b + 4 b a + 4 \cdot - 1 b \cdot b}{(a + b) (a - b)}$

Step 8.7

Simplify each term.

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Step 8.7.1

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

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Step 8.7.1.1

Move $b$ .

$\frac{5 a^{2} + 6 a b + 4 b a + 4 \cdot - 1 (b \cdot b)}{(a + b) (a - b)}$

Step 8.7.1.2

Multiply $b$ by $b$ .

$\frac{5 a^{2} + 6 a b + 4 b a + 4 \cdot - 1 b^{2}}{(a + b) (a - b)}$

Step 8.7.2

Multiply $4$ by $- 1$ .

$\frac{5 a^{2} + 6 a b + 4 b a - 4 b^{2}}{(a + b) (a - b)}$

Step 8.8

Add $6 a b$ and $4 b a$ .

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Step 8.8.1

Move $b$ .

$\frac{5 a^{2} + 6 a b + 4 a b - 4 b^{2}}{(a + b) (a - b)}$

Step 8.8.2

Add $6 a b$ and $4 a b$ .

$\frac{5 a^{2} + 10 a b - 4 b^{2}}{(a + b) (a - b)}$