Basic Math Examples

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

Step 1

Simplify the denominator.

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Step 1.1

Rewrite $1$ as $1^{2}$ .

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

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 $a = a$ and $b = 1$ .

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

Step 2

To write $\frac{2 a}{(a + 1) (a - 1)}$ as a fraction with a common denominator, multiply by $\frac{a^{2 + a}}{a^{2 + a}}$ .

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

Step 3

To write $\frac{1}{a^{2 + a}}$ as a fraction with a common denominator, multiply by $\frac{(a + 1) (a - 1)}{(a + 1) (a - 1)}$ .

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

Step 4

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

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

Reorder the factors of $(a + 1) (a - 1) a^{2 + a}$ .

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

Step 4.4

Reorder the factors of $a^{2 + a} ((a + 1) (a - 1))$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Multiply $a$ by $a^{2 + a}$ by adding the exponents.

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Step 6.1.1

Move $a^{2 + a}$ .

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

Step 6.1.2

Multiply $a^{2 + a}$ by $a$ .

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Step 6.1.2.1

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

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

Step 6.1.2.2

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

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

Step 6.1.3

Add $2$ and $1$ .

$\frac{2 a^{a + 3} + (a + 1) (a - 1)}{a^{2 + a} (a + 1) (a - 1)}$

Step 6.2

Expand $(a + 1) (a - 1)$ using the FOIL Method.

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Step 6.2.1

Apply the distributive property.

$\frac{2 a^{a + 3} + a (a - 1) + 1 (a - 1)}{a^{2 + a} (a + 1) (a - 1)}$

Step 6.2.2

Apply the distributive property.

$\frac{2 a^{a + 3} + a \cdot a + a \cdot - 1 + 1 (a - 1)}{a^{2 + a} (a + 1) (a - 1)}$

Step 6.2.3

Apply the distributive property.

$\frac{2 a^{a + 3} + a \cdot a + a \cdot - 1 + 1 a + 1 \cdot - 1}{a^{2 + a} (a + 1) (a - 1)}$

Step 6.3

Simplify and combine like terms.

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Step 6.3.1

Simplify each term.

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Step 6.3.1.1

Multiply $a$ by $a$ .

$\frac{2 a^{a + 3} + a^{2} + a \cdot - 1 + 1 a + 1 \cdot - 1}{a^{2 + a} (a + 1) (a - 1)}$

Step 6.3.1.2

Move $- 1$ to the left of $a$ .

$\frac{2 a^{a + 3} + a^{2} - 1 \cdot a + 1 a + 1 \cdot - 1}{a^{2 + a} (a + 1) (a - 1)}$

Step 6.3.1.3

Rewrite $- 1 a$ as $- a$ .

$\frac{2 a^{a + 3} + a^{2} - a + 1 a + 1 \cdot - 1}{a^{2 + a} (a + 1) (a - 1)}$

Step 6.3.1.4

Multiply $a$ by $1$ .

$\frac{2 a^{a + 3} + a^{2} - a + a + 1 \cdot - 1}{a^{2 + a} (a + 1) (a - 1)}$

Step 6.3.1.5

Multiply $- 1$ by $1$ .

$\frac{2 a^{a + 3} + a^{2} - a + a - 1}{a^{2 + a} (a + 1) (a - 1)}$

Step 6.3.2

Add $- a$ and $a$ .

$\frac{2 a^{a + 3} + a^{2} + 0 - 1}{a^{2 + a} (a + 1) (a - 1)}$

Step 6.3.3

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

$\frac{2 a^{a + 3} + a^{2} - 1}{a^{2 + a} (a + 1) (a - 1)}$

Step 6.4

Reorder terms.

$\frac{a^{2} + 2 a^{a + 3} - 1}{a^{2 + a} (a + 1) (a - 1)}$