Algebra Examples

$a^{2} - \frac{a^{4} + 1}{a^{2} - 1} + 1$

Step 1

Find the common denominator.

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Step 1.1

Write $a^{2}$ as a fraction with denominator $1$ .

$\frac{a^{2}}{1} - \frac{a^{4} + 1}{a^{2} - 1} + 1$

Step 1.2

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

$\frac{a^{2}}{1} \cdot \frac{a^{2} - 1}{a^{2} - 1} - \frac{a^{4} + 1}{a^{2} - 1} + 1$

Step 1.3

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

$\frac{a^{2} (a^{2} - 1)}{a^{2} - 1} - \frac{a^{4} + 1}{a^{2} - 1} + 1$

Step 1.4

Write $1$ as a fraction with denominator $1$ .

$\frac{a^{2} (a^{2} - 1)}{a^{2} - 1} - \frac{a^{4} + 1}{a^{2} - 1} + \frac{1}{1}$

Step 1.5

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

$\frac{a^{2} (a^{2} - 1)}{a^{2} - 1} - \frac{a^{4} + 1}{a^{2} - 1} + \frac{1}{1} \cdot \frac{a^{2} - 1}{a^{2} - 1}$

Step 1.6

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

$\frac{a^{2} (a^{2} - 1)}{a^{2} - 1} - \frac{a^{4} + 1}{a^{2} - 1} + \frac{a^{2} - 1}{a^{2} - 1}$

Step 2

Simplify terms.

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Step 2.1

Combine the numerators over the common denominator.

$\frac{a^{2} (a^{2} - 1) - (a^{4} + 1) + a^{2} - 1}{a^{2} - 1}$

Step 2.2

Simplify each term.

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Step 2.2.1

Apply the distributive property.

$\frac{a^{2} a^{2} + a^{2} \cdot - 1 - (a^{4} + 1) + a^{2} - 1}{a^{2} - 1}$

Step 2.2.2

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

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Step 2.2.2.1

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

$\frac{a^{2 + 2} + a^{2} \cdot - 1 - (a^{4} + 1) + a^{2} - 1}{a^{2} - 1}$

Step 2.2.2.2

Add $2$ and $2$ .

$\frac{a^{4} + a^{2} \cdot - 1 - (a^{4} + 1) + a^{2} - 1}{a^{2} - 1}$

Step 2.2.3

Move $- 1$ to the left of $a^{2}$ .

$\frac{a^{4} - 1 \cdot a^{2} - (a^{4} + 1) + a^{2} - 1}{a^{2} - 1}$

Step 2.2.4

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

$\frac{a^{4} - a^{2} - (a^{4} + 1) + a^{2} - 1}{a^{2} - 1}$

Step 2.2.5

Apply the distributive property.

$\frac{a^{4} - a^{2} - a^{4} - 1 \cdot 1 + a^{2} - 1}{a^{2} - 1}$

Step 2.2.6

Multiply $- 1$ by $1$ .

$\frac{a^{4} - a^{2} - a^{4} - 1 + a^{2} - 1}{a^{2} - 1}$

Step 2.3

Simplify by adding terms.

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Step 2.3.1

Combine the opposite terms in $a^{4} - a^{2} - a^{4} - 1 + a^{2} - 1$ .

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Step 2.3.1.1

Subtract $a^{4}$ from $a^{4}$ .

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

Step 2.3.1.2

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

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

Step 2.3.1.3

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

$\frac{0 - 1 - 1}{a^{2} - 1}$

Step 2.3.1.4

Subtract $1$ from $0$ .

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

Step 2.3.2

Subtract $1$ from $- 1$ .

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

Step 3

Simplify the denominator.

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Step 3.1

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

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

Step 3.2

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 $b = 1$ .

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

Step 4

Move the negative in front of the fraction.

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