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 $a$ .

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