Algebra Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

Write each expression with a common denominator of $(b - a) {(b + a)}^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{2 b}{{(b + a)}^{2}}$ by $\frac{b - a}{b - a}$ .

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

Step 4.2

Multiply $\frac{2 a}{(b + a) (b - a)}$ by $\frac{b + a}{b + a}$ .

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

Step 4.3

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

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

Step 4.4

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

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

Step 4.5

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

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

Step 4.6

Add $1$ and $1$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

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

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Step 6.1.1

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

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

Step 6.1.2

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

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

Step 6.1.3

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

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

Step 6.2

Apply the distributive property.

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

Step 6.3

Multiply $b$ by $b$ .

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

Step 6.4

Rewrite using the commutative property of multiplication.

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

Step 6.5

Apply the distributive property.

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

Step 6.6

Multiply $a$ by $a$ .

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

Step 6.7

Add $- b a$ and $a b$ .

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Step 6.7.1

Move $b$ .

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

Step 6.7.2

Add $- a b$ and $a b$ .

$\frac{2 (b^{2} + 0 + a^{2})}{{(b + a)}^{2} (b - a)}$

Step 6.8

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

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