Basic Math Examples

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

Step 1

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

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

Step 2

Expand $(a + \frac{1}{a} - 1) (a + \frac{1}{a} - 1)$ by multiplying each term in the first expression by each term in the second expression.

$a \cdot a + a \frac{1}{a} + a \cdot - 1 + \frac{1}{a} a + \frac{1}{a} \cdot \frac{1}{a} + \frac{1}{a} \cdot - 1 - 1 a - 1 \frac{1}{a} - 1 \cdot - 1$

Step 3

Simplify terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Multiply $a$ by $a$ .

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

Step 3.1.2

Cancel the common factor of $a$ .

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Step 3.1.2.1

Cancel the common factor.

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

Step 3.1.2.2

Rewrite the expression.

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

Step 3.1.3

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

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

Step 3.1.4

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

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

Step 3.1.5

Cancel the common factor of $a$ .

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Step 3.1.5.1

Cancel the common factor.

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

Step 3.1.5.2

Rewrite the expression.

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

Step 3.1.6

Multiply $\frac{1}{a} \cdot \frac{1}{a}$ .

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Step 3.1.6.1

Multiply $\frac{1}{a}$ by $\frac{1}{a}$ .

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

Step 3.1.6.2

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

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

Step 3.1.6.3

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

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

Step 3.1.6.4

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

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

Step 3.1.6.5

Add $1$ and $1$ .

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

Step 3.1.7

Combine $\frac{1}{a}$ and $- 1$ .

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

Step 3.1.8

Move the negative in front of the fraction.

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

Step 3.1.9

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

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

Step 3.1.10

Rewrite $- 1 \frac{1}{a}$ as $- \frac{1}{a}$ .

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

Step 3.1.11

Multiply $- 1$ by $- 1$ .

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

Step 3.2

Simplify terms.

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Step 3.2.1

Combine the numerators over the common denominator.

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

Step 3.2.2

Simplify the expression.

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Step 3.2.2.1

Subtract $1$ from $- 1$ .

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

Step 3.2.2.2

Move the negative in front of the fraction.

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

Step 3.2.2.3

Add $1$ and $1$ .

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

Step 3.2.3

Subtract $a$ from $- a$ .

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

Step 3.2.4

Add $2$ and $1$ .

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