Basic Math Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

Combine the numerators over the common denominator.

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

Step 1.3

Apply the product rule to $\frac{s + 1}{s}$ .

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

Step 1.4

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

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

Step 1.5

Combine the numerators over the common denominator.

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

Step 1.6

Apply the product rule to $\frac{s - 1}{s}$ .

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

Step 2

Combine the numerators over the common denominator.

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

Step 3

Simplify each term.

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Step 3.1

Rewrite ${(s + 1)}^{2}$ as $(s + 1) (s + 1)$ .

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

Step 3.2

Expand $(s + 1) (s + 1)$ using the FOIL Method.

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Step 3.2.1

Apply the distributive property.

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

Step 3.2.2

Apply the distributive property.

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

Step 3.2.3

Apply the distributive property.

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

Step 3.3

Simplify and combine like terms.

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Step 3.3.1

Simplify each term.

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Step 3.3.1.1

Multiply $s$ by $s$ .

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

Step 3.3.1.2

Multiply $s$ by $1$ .

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

Step 3.3.1.3

Multiply $s$ by $1$ .

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

Step 3.3.1.4

Multiply $1$ by $1$ .

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

Step 3.3.2

Add $s$ and $s$ .

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

Step 3.4

Rewrite ${(s - 1)}^{2}$ as $(s - 1) (s - 1)$ .

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

Step 3.5

Expand $(s - 1) (s - 1)$ using the FOIL Method.

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Step 3.5.1

Apply the distributive property.

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

Step 3.5.2

Apply the distributive property.

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

Step 3.5.3

Apply the distributive property.

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

Step 3.6

Simplify and combine like terms.

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Step 3.6.1

Simplify each term.

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Step 3.6.1.1

Multiply $s$ by $s$ .

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

Step 3.6.1.2

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

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

Step 3.6.1.3

Rewrite $- 1 s$ as $- s$ .

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

Step 3.6.1.4

Rewrite $- 1 s$ as $- s$ .

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

Step 3.6.1.5

Multiply $- 1$ by $- 1$ .

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

Step 3.6.2

Subtract $s$ from $- s$ .

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

Step 3.7

Apply the distributive property.

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

Step 3.8

Simplify.

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Step 3.8.1

Multiply $- 2$ by $- 1$ .

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

Step 3.8.2

Multiply $- 1$ by $1$ .

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

Step 4

Simplify terms.

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Step 4.1

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

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Step 4.1.1

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

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

Step 4.1.2

Add $2 s + 1$ and $0$ .

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

Step 4.1.3

Subtract $1$ from $1$ .

$\frac{2 s + 2 s + 0}{s^{2}}$

Step 4.1.4

Add $2 s + 2 s$ and $0$ .

$\frac{2 s + 2 s}{s^{2}}$

Step 4.2

Add $2 s$ and $2 s$ .

$\frac{4 s}{s^{2}}$

Step 4.3

Cancel the common factor of $s$ and $s^{2}$ .

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Step 4.3.1

Factor $s$ out of $4 s$ .

$\frac{s \cdot 4}{s^{2}}$

Step 4.3.2

Cancel the common factors.

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Step 4.3.2.1

Factor $s$ out of $s^{2}$ .

$\frac{s \cdot 4}{s \cdot s}$

Step 4.3.2.2

Cancel the common factor.

$\frac{s \cdot 4}{s \cdot s}$

Step 4.3.2.3

Rewrite the expression.

$\frac{4}{s}$