Basic Math Examples

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

Step 1

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

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

Step 2

Write each expression with a common denominator of ${(a^{2} + 5)}^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.1

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

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

Step 2.2

Raise $a^{2} + 5$ to the power of $1$ .

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

Step 2.3

Raise $a^{2} + 5$ to the power of $1$ .

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

Step 2.4

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

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

Step 2.5

Add $1$ and $1$ .

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

Step 3

Combine the numerators over the common denominator.

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

Step 4

Simplify the numerator.

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Step 4.1

Apply the distributive property.

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

Step 4.2

Multiply $- 1$ by $- 5$ .

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

Step 4.3

Subtract $a^{2}$ from $a^{2}$ .

$\frac{0 + 5 + 5}{{(a^{2} + 5)}^{2}} + \frac{1}{a^{2} - 5}$

Step 4.4

Add $0$ and $5$ .

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

Step 4.5

Add $5$ and $5$ .

$\frac{10}{{(a^{2} + 5)}^{2}} + \frac{1}{a^{2} - 5}$

Step 5

To write $\frac{10}{{(a^{2} + 5)}^{2}}$ as a fraction with a common denominator, multiply by $\frac{a^{2} - 5}{a^{2} - 5}$ .

$\frac{10}{{(a^{2} + 5)}^{2}} \cdot \frac{a^{2} - 5}{a^{2} - 5} + \frac{1}{a^{2} - 5}$

Step 6

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

$\frac{10}{{(a^{2} + 5)}^{2}} \cdot \frac{a^{2} - 5}{a^{2} - 5} + \frac{1}{a^{2} - 5} \cdot \frac{{(a^{2} + 5)}^{2}}{{(a^{2} + 5)}^{2}}$

Step 7

Write each expression with a common denominator of $(a^{2} - 5) {(a^{2} + 5)}^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 7.1

Multiply $\frac{10}{{(a^{2} + 5)}^{2}}$ by $\frac{a^{2} - 5}{a^{2} - 5}$ .

$\frac{10 (a^{2} - 5)}{{(a^{2} + 5)}^{2} (a^{2} - 5)} + \frac{1}{a^{2} - 5} \cdot \frac{{(a^{2} + 5)}^{2}}{{(a^{2} + 5)}^{2}}$

Step 7.2

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

$\frac{10 (a^{2} - 5)}{{(a^{2} + 5)}^{2} (a^{2} - 5)} + \frac{{(a^{2} + 5)}^{2}}{(a^{2} - 5) {(a^{2} + 5)}^{2}}$

Step 7.3

Reorder the factors of $(a^{2} - 5) {(a^{2} + 5)}^{2}$ .

$\frac{10 (a^{2} - 5)}{{(a^{2} + 5)}^{2} (a^{2} - 5)} + \frac{{(a^{2} + 5)}^{2}}{{(a^{2} + 5)}^{2} (a^{2} - 5)}$

Step 8

Combine the numerators over the common denominator.

$\frac{10 (a^{2} - 5) + {(a^{2} + 5)}^{2}}{{(a^{2} + 5)}^{2} (a^{2} - 5)}$

Step 9

Simplify the numerator.

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Step 9.1

Apply the distributive property.

$\frac{10 a^{2} + 10 \cdot - 5 + {(a^{2} + 5)}^{2}}{{(a^{2} + 5)}^{2} (a^{2} - 5)}$

Step 9.2

Multiply $10$ by $- 5$ .

$\frac{10 a^{2} - 50 + {(a^{2} + 5)}^{2}}{{(a^{2} + 5)}^{2} (a^{2} - 5)}$

Step 9.3

Rewrite ${(a^{2} + 5)}^{2}$ as $(a^{2} + 5) (a^{2} + 5)$ .

$\frac{10 a^{2} - 50 + (a^{2} + 5) (a^{2} + 5)}{{(a^{2} + 5)}^{2} (a^{2} - 5)}$

Step 9.4

Expand $(a^{2} + 5) (a^{2} + 5)$ using the FOIL Method.

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Step 9.4.1

Apply the distributive property.

$\frac{10 a^{2} - 50 + a^{2} (a^{2} + 5) + 5 (a^{2} + 5)}{{(a^{2} + 5)}^{2} (a^{2} - 5)}$

Step 9.4.2

Apply the distributive property.

$\frac{10 a^{2} - 50 + a^{2} a^{2} + a^{2} \cdot 5 + 5 (a^{2} + 5)}{{(a^{2} + 5)}^{2} (a^{2} - 5)}$

Step 9.4.3

Apply the distributive property.

$\frac{10 a^{2} - 50 + a^{2} a^{2} + a^{2} \cdot 5 + 5 a^{2} + 5 \cdot 5}{{(a^{2} + 5)}^{2} (a^{2} - 5)}$

Step 9.5

Simplify and combine like terms.

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Step 9.5.1

Simplify each term.

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Step 9.5.1.1

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

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Step 9.5.1.1.1

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

$\frac{10 a^{2} - 50 + a^{2 + 2} + a^{2} \cdot 5 + 5 a^{2} + 5 \cdot 5}{{(a^{2} + 5)}^{2} (a^{2} - 5)}$

Step 9.5.1.1.2

Add $2$ and $2$ .

$\frac{10 a^{2} - 50 + a^{4} + a^{2} \cdot 5 + 5 a^{2} + 5 \cdot 5}{{(a^{2} + 5)}^{2} (a^{2} - 5)}$

Step 9.5.1.2

Move $5$ to the left of $a^{2}$ .

$\frac{10 a^{2} - 50 + a^{4} + 5 \cdot a^{2} + 5 a^{2} + 5 \cdot 5}{{(a^{2} + 5)}^{2} (a^{2} - 5)}$

Step 9.5.1.3

Multiply $5$ by $5$ .

$\frac{10 a^{2} - 50 + a^{4} + 5 a^{2} + 5 a^{2} + 25}{{(a^{2} + 5)}^{2} (a^{2} - 5)}$

Step 9.5.2

Add $5 a^{2}$ and $5 a^{2}$ .

$\frac{10 a^{2} - 50 + a^{4} + 10 a^{2} + 25}{{(a^{2} + 5)}^{2} (a^{2} - 5)}$

Step 9.6

Add $10 a^{2}$ and $10 a^{2}$ .

$\frac{20 a^{2} - 50 + a^{4} + 25}{{(a^{2} + 5)}^{2} (a^{2} - 5)}$

Step 9.7

Add $- 50$ and $25$ .

$\frac{20 a^{2} + a^{4} - 25}{{(a^{2} + 5)}^{2} (a^{2} - 5)}$

Step 9.8

Reorder terms.

$\frac{a^{4} + 20 a^{2} - 25}{{(a^{2} + 5)}^{2} (a^{2} - 5)}$