Basic Math Examples

$\frac{5}{a^{2} + 2 a + 1} + \frac{8}{a^{2} - 1}$

Step 1

Simplify each term.

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Step 1.1

Factor using the perfect square rule.

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Step 1.1.1

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

$\frac{5}{a^{2} + 2 a + 1^{2}} + \frac{8}{a^{2} - 1}$

Step 1.1.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 a = 2 \cdot a \cdot 1$

Step 1.1.3

Rewrite the polynomial.

$\frac{5}{a^{2} + 2 \cdot a \cdot 1 + 1^{2}} + \frac{8}{a^{2} - 1}$

Step 1.1.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = a$ and $b = 1$ .

$\frac{5}{{(a + 1)}^{2}} + \frac{8}{a^{2} - 1}$

Step 1.2

Simplify the denominator.

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Step 1.2.1

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

$\frac{5}{{(a + 1)}^{2}} + \frac{8}{a^{2} - 1^{2}}$

Step 1.2.2

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 $b = 1$ .

$\frac{5}{{(a + 1)}^{2}} + \frac{8}{(a + 1) (a - 1)}$

Step 2

To write $\frac{5}{{(a + 1)}^{2}}$ as a fraction with a common denominator, multiply by $\frac{a - 1}{a - 1}$ .

$\frac{5}{{(a + 1)}^{2}} \cdot \frac{a - 1}{a - 1} + \frac{8}{(a + 1) (a - 1)}$

Step 3

To write $\frac{8}{(a + 1) (a - 1)}$ as a fraction with a common denominator, multiply by $\frac{a + 1}{a + 1}$ .

$\frac{5}{{(a + 1)}^{2}} \cdot \frac{a - 1}{a - 1} + \frac{8}{(a + 1) (a - 1)} \cdot \frac{a + 1}{a + 1}$

Step 4

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

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Step 4.1

Multiply $\frac{5}{{(a + 1)}^{2}}$ by $\frac{a - 1}{a - 1}$ .

$\frac{5 (a - 1)}{{(a + 1)}^{2} (a - 1)} + \frac{8}{(a + 1) (a - 1)} \cdot \frac{a + 1}{a + 1}$

Step 4.2

Multiply $\frac{8}{(a + 1) (a - 1)}$ by $\frac{a + 1}{a + 1}$ .

$\frac{5 (a - 1)}{{(a + 1)}^{2} (a - 1)} + \frac{8 (a + 1)}{(a + 1) (a - 1) (a + 1)}$

Step 4.3

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

$\frac{5 (a - 1)}{{(a + 1)}^{2} (a - 1)} + \frac{8 (a + 1)}{{(a + 1)}^{1} (a + 1) (a - 1)}$

Step 4.4

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

$\frac{5 (a - 1)}{{(a + 1)}^{2} (a - 1)} + \frac{8 (a + 1)}{{(a + 1)}^{1} {(a + 1)}^{1} (a - 1)}$

Step 4.5

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

$\frac{5 (a - 1)}{{(a + 1)}^{2} (a - 1)} + \frac{8 (a + 1)}{{(a + 1)}^{1 + 1} (a - 1)}$

Step 4.6

Add $1$ and $1$ .

$\frac{5 (a - 1)}{{(a + 1)}^{2} (a - 1)} + \frac{8 (a + 1)}{{(a + 1)}^{2} (a - 1)}$

Step 5

Combine the numerators over the common denominator.

$\frac{5 (a - 1) + 8 (a + 1)}{{(a + 1)}^{2} (a - 1)}$

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

$\frac{5 a + 5 \cdot - 1 + 8 (a + 1)}{{(a + 1)}^{2} (a - 1)}$

Step 6.2

Multiply $5$ by $- 1$ .

$\frac{5 a - 5 + 8 (a + 1)}{{(a + 1)}^{2} (a - 1)}$

Step 6.3

Apply the distributive property.

$\frac{5 a - 5 + 8 a + 8 \cdot 1}{{(a + 1)}^{2} (a - 1)}$

Step 6.4

Multiply $8$ by $1$ .

$\frac{5 a - 5 + 8 a + 8}{{(a + 1)}^{2} (a - 1)}$

Step 6.5

Add $5 a$ and $8 a$ .

$\frac{13 a - 5 + 8}{{(a + 1)}^{2} (a - 1)}$

Step 6.6

Add $- 5$ and $8$ .

$\frac{13 a + 3}{{(a + 1)}^{2} (a - 1)}$