Basic Math Examples

$\frac{a}{1 - a^{2}} + \frac{a}{1 + a^{2}}$

Step 1

Simplify the denominator.

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Step 1.1

Rewrite

$1$ as

$1^{2}$ .

$\frac{a}{1^{2} - a^{2}} + \frac{a}{1 + a^{2}}$

Step 1.2

Since both terms are perfect squares, factor using the difference of squares formula,

$a^{2} - b^{2} = (a + b) (a - b)$ where

$a = 1$ and

$b = a$ .

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

Step 2

To write

$\frac{a}{(1 + a) (1 - a)}$ as a fraction with a common denominator, multiply by

$\frac{1 + a^{2}}{1 + a^{2}}$ .

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

Step 3

To write

$\frac{a}{1 + a^{2}}$ as a fraction with a common denominator, multiply by

$\frac{(1 + a) (1 - a)}{(1 + a) (1 - a)}$ .

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

Step 4

Write each expression with a common denominator of

$(1 + a) (1 - a) (1 + a^{2})$ , by multiplying each by an appropriate factor of

$1$ .

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Step 4.1

Multiply

$\frac{a}{(1 + a) (1 - a)}$ by

$\frac{1 + a^{2}}{1 + a^{2}}$ .

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

Step 4.2

Multiply

$\frac{a}{1 + a^{2}}$ by

$\frac{(1 + a) (1 - a)}{(1 + a) (1 - a)}$ .

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

Step 4.3

Reorder the factors of

$(1 + a) (1 - a) (1 + a^{2})$ .

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

Step 4.4

Reorder the factors of

$(1 + a^{2}) ((1 + a) (1 - a))$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Factor

$a$ out of

$a (1 + a^{2}) + a ((1 + a) (1 - a))$ .

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

Step 6.2

Let

$u = 1$ . Substitute

$u$ for all occurrences of

$1$ .

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Step 6.2.1

Subtract

$a^{2}$ from

$a^{2}$ .

$\frac{a (u + u^{2} + 0)}{(1 + a) (1 + a^{2}) (1 - a)}$

Step 6.2.2

Add

$u + u^{2}$ and

$0$ .

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

Step 6.3

Factor

$u$ out of

$u + u^{2}$ .

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Step 6.3.1

Raise

$u$ to the power of

$1$ .

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

Step 6.3.2

Factor

$u$ out of

$u^{1}$ .

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

Step 6.3.3

Factor

$u$ out of

$u^{2}$ .

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

Step 6.3.4

Factor

$u$ out of

$u \cdot 1 + u \cdot u$ .

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

Step 6.4

Replace all occurrences of

$u$ with

$1$ .

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

Step 6.5

Add

$1$ and

$1$ .

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

Step 6.6

Multiply

$2$ by

$1$ .

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

Step 7

Move

$2$ to the left of

$a$ .

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