Basic Math Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

Expand $(1 - \frac{1}{y}) (1 - \frac{1}{y})$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

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

Step 1.2.2

Apply the distributive property.

$1 + \frac{1}{y} - (1 \cdot 1 + 1 (- \frac{1}{y}) - \frac{1}{y} (1 - \frac{1}{y}))$

Step 1.2.3

Apply the distributive property.

$1 + \frac{1}{y} - (1 \cdot 1 + 1 (- \frac{1}{y}) - \frac{1}{y} \cdot 1 - \frac{1}{y} (- \frac{1}{y}))$

Step 1.3

Simplify and combine like terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Multiply $1$ by $1$ .

$1 + \frac{1}{y} - (1 + 1 (- \frac{1}{y}) - \frac{1}{y} \cdot 1 - \frac{1}{y} (- \frac{1}{y}))$

Step 1.3.1.2

Multiply $- \frac{1}{y}$ by $1$ .

$1 + \frac{1}{y} - (1 - \frac{1}{y} - \frac{1}{y} \cdot 1 - \frac{1}{y} (- \frac{1}{y}))$

Step 1.3.1.3

Multiply $- 1$ by $1$ .

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

Step 1.3.1.4

Multiply $- \frac{1}{y} (- \frac{1}{y})$ .

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Step 1.3.1.4.1

Multiply $- 1$ by $- 1$ .

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

Step 1.3.1.4.2

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

$1 + \frac{1}{y} - (1 - \frac{1}{y} - \frac{1}{y} + \frac{1}{y} \cdot \frac{1}{y})$

Step 1.3.1.4.3

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

$1 + \frac{1}{y} - (1 - \frac{1}{y} - \frac{1}{y} + \frac{1}{y \cdot y})$

Step 1.3.1.4.4

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

$1 + \frac{1}{y} - (1 - \frac{1}{y} - \frac{1}{y} + \frac{1}{y^{1} y})$

Step 1.3.1.4.5

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

$1 + \frac{1}{y} - (1 - \frac{1}{y} - \frac{1}{y} + \frac{1}{y^{1} y^{1}})$

Step 1.3.1.4.6

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

$1 + \frac{1}{y} - (1 - \frac{1}{y} - \frac{1}{y} + \frac{1}{y^{1 + 1}})$

Step 1.3.1.4.7

Add $1$ and $1$ .

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

Step 1.3.2

Subtract $\frac{1}{y}$ from $- \frac{1}{y}$ .

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

Step 1.4

Simplify each term.

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Step 1.4.1

Combine $- 2$ and $\frac{1}{y}$ .

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

Step 1.4.2

Move the negative in front of the fraction.

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

Step 1.5

Apply the distributive property.

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

Step 1.6

Simplify.

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Step 1.6.1

Multiply $- 1$ by $1$ .

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

Step 1.6.2

Multiply $- - \frac{2}{y}$ .

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Step 1.6.2.1

Multiply $- 1$ by $- 1$ .

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

Step 1.6.2.2

Multiply $\frac{2}{y}$ by $1$ .

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

Step 2

Simplify terms.

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Step 2.1

Combine the opposite terms in $1 + \frac{1}{y} - 1 + \frac{2}{y} - \frac{1}{y^{2}}$ .

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Step 2.1.1

Subtract $1$ from $1$ .

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

Step 2.1.2

Add $\frac{1}{y}$ and $0$ .

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

Step 2.2

Combine the numerators over the common denominator.

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

Step 2.3

Simplify the expression.

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Step 2.3.1

Add $1$ and $2$ .

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

Step 2.3.2

Move the negative in front of the fraction.

$\frac{3}{y} - \frac{1}{y^{2}}$