Basic Math Examples

${(\frac{2}{s} - \frac{1}{s^{3}})}^{2}$

Step 1

Rewrite ${(\frac{2}{s} - \frac{1}{s^{3}})}^{2}$ as $(\frac{2}{s} - \frac{1}{s^{3}}) (\frac{2}{s} - \frac{1}{s^{3}})$ .

$(\frac{2}{s} - \frac{1}{s^{3}}) (\frac{2}{s} - \frac{1}{s^{3}})$

Step 2

Expand $(\frac{2}{s} - \frac{1}{s^{3}}) (\frac{2}{s} - \frac{1}{s^{3}})$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$\frac{2}{s} (\frac{2}{s} - \frac{1}{s^{3}}) - \frac{1}{s^{3}} (\frac{2}{s} - \frac{1}{s^{3}})$

Step 2.2

Apply the distributive property.

$\frac{2}{s} \cdot \frac{2}{s} + \frac{2}{s} (- \frac{1}{s^{3}}) - \frac{1}{s^{3}} (\frac{2}{s} - \frac{1}{s^{3}})$

Step 2.3

Apply the distributive property.

$\frac{2}{s} \cdot \frac{2}{s} + \frac{2}{s} (- \frac{1}{s^{3}}) - \frac{1}{s^{3}} \cdot \frac{2}{s} - \frac{1}{s^{3}} (- \frac{1}{s^{3}})$

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Multiply $\frac{2}{s} \cdot \frac{2}{s}$ .

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Step 3.1.1.1

Multiply $\frac{2}{s}$ by $\frac{2}{s}$ .

$\frac{2 \cdot 2}{s \cdot s} + \frac{2}{s} (- \frac{1}{s^{3}}) - \frac{1}{s^{3}} \cdot \frac{2}{s} - \frac{1}{s^{3}} (- \frac{1}{s^{3}})$

Step 3.1.1.2

Multiply $2$ by $2$ .

$\frac{4}{s \cdot s} + \frac{2}{s} (- \frac{1}{s^{3}}) - \frac{1}{s^{3}} \cdot \frac{2}{s} - \frac{1}{s^{3}} (- \frac{1}{s^{3}})$

Step 3.1.1.3

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

$\frac{4}{s^{1} s} + \frac{2}{s} (- \frac{1}{s^{3}}) - \frac{1}{s^{3}} \cdot \frac{2}{s} - \frac{1}{s^{3}} (- \frac{1}{s^{3}})$

Step 3.1.1.4

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

$\frac{4}{s^{1} s^{1}} + \frac{2}{s} (- \frac{1}{s^{3}}) - \frac{1}{s^{3}} \cdot \frac{2}{s} - \frac{1}{s^{3}} (- \frac{1}{s^{3}})$

Step 3.1.1.5

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

$\frac{4}{s^{1 + 1}} + \frac{2}{s} (- \frac{1}{s^{3}}) - \frac{1}{s^{3}} \cdot \frac{2}{s} - \frac{1}{s^{3}} (- \frac{1}{s^{3}})$

Step 3.1.1.6

Add $1$ and $1$ .

$\frac{4}{s^{2}} + \frac{2}{s} (- \frac{1}{s^{3}}) - \frac{1}{s^{3}} \cdot \frac{2}{s} - \frac{1}{s^{3}} (- \frac{1}{s^{3}})$

Step 3.1.2

Rewrite using the commutative property of multiplication.

$\frac{4}{s^{2}} - \frac{2}{s} \cdot \frac{1}{s^{3}} - \frac{1}{s^{3}} \cdot \frac{2}{s} - \frac{1}{s^{3}} (- \frac{1}{s^{3}})$

Step 3.1.3

Multiply $- \frac{2}{s} \cdot \frac{1}{s^{3}}$ .

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Step 3.1.3.1

Multiply $\frac{1}{s^{3}}$ by $\frac{2}{s}$ .

$\frac{4}{s^{2}} - \frac{2}{s^{3} s} - \frac{1}{s^{3}} \cdot \frac{2}{s} - \frac{1}{s^{3}} (- \frac{1}{s^{3}})$

Step 3.1.3.2

Multiply $s^{3}$ by $s$ by adding the exponents.

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Step 3.1.3.2.1

Multiply $s^{3}$ by $s$ .

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Step 3.1.3.2.1.1

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

$\frac{4}{s^{2}} - \frac{2}{s^{3} s^{1}} - \frac{1}{s^{3}} \cdot \frac{2}{s} - \frac{1}{s^{3}} (- \frac{1}{s^{3}})$

Step 3.1.3.2.1.2

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

$\frac{4}{s^{2}} - \frac{2}{s^{3 + 1}} - \frac{1}{s^{3}} \cdot \frac{2}{s} - \frac{1}{s^{3}} (- \frac{1}{s^{3}})$

Step 3.1.3.2.2

Add $3$ and $1$ .

$\frac{4}{s^{2}} - \frac{2}{s^{4}} - \frac{1}{s^{3}} \cdot \frac{2}{s} - \frac{1}{s^{3}} (- \frac{1}{s^{3}})$

Step 3.1.4

Multiply $- \frac{1}{s^{3}} \cdot \frac{2}{s}$ .

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Step 3.1.4.1

Multiply $\frac{2}{s}$ by $\frac{1}{s^{3}}$ .

$\frac{4}{s^{2}} - \frac{2}{s^{4}} - \frac{2}{s \cdot s^{3}} - \frac{1}{s^{3}} (- \frac{1}{s^{3}})$

Step 3.1.4.2

Multiply $s$ by $s^{3}$ by adding the exponents.

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Step 3.1.4.2.1

Multiply $s$ by $s^{3}$ .

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Step 3.1.4.2.1.1

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

$\frac{4}{s^{2}} - \frac{2}{s^{4}} - \frac{2}{s^{1} s^{3}} - \frac{1}{s^{3}} (- \frac{1}{s^{3}})$

Step 3.1.4.2.1.2

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

$\frac{4}{s^{2}} - \frac{2}{s^{4}} - \frac{2}{s^{1 + 3}} - \frac{1}{s^{3}} (- \frac{1}{s^{3}})$

Step 3.1.4.2.2

Add $1$ and $3$ .

$\frac{4}{s^{2}} - \frac{2}{s^{4}} - \frac{2}{s^{4}} - \frac{1}{s^{3}} (- \frac{1}{s^{3}})$

Step 3.1.5

Multiply $- \frac{1}{s^{3}} (- \frac{1}{s^{3}})$ .

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Step 3.1.5.1

Multiply $- 1$ by $- 1$ .

$\frac{4}{s^{2}} - \frac{2}{s^{4}} - \frac{2}{s^{4}} + 1 \frac{1}{s^{3}} \frac{1}{s^{3}}$

Step 3.1.5.2

Multiply $\frac{1}{s^{3}}$ by $1$ .

$\frac{4}{s^{2}} - \frac{2}{s^{4}} - \frac{2}{s^{4}} + \frac{1}{s^{3}} \cdot \frac{1}{s^{3}}$

Step 3.1.5.3

Multiply $\frac{1}{s^{3}}$ by $\frac{1}{s^{3}}$ .

$\frac{4}{s^{2}} - \frac{2}{s^{4}} - \frac{2}{s^{4}} + \frac{1}{s^{3} s^{3}}$

Step 3.1.5.4

Multiply $s^{3}$ by $s^{3}$ by adding the exponents.

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Step 3.1.5.4.1

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

$\frac{4}{s^{2}} - \frac{2}{s^{4}} - \frac{2}{s^{4}} + \frac{1}{s^{3 + 3}}$

Step 3.1.5.4.2

Add $3$ and $3$ .

$\frac{4}{s^{2}} - \frac{2}{s^{4}} - \frac{2}{s^{4}} + \frac{1}{s^{6}}$

Step 3.2

Subtract $\frac{2}{s^{4}}$ from $- \frac{2}{s^{4}}$ .

$\frac{4}{s^{2}} - 2 \frac{2}{s^{4}} + \frac{1}{s^{6}}$

Step 4

Simplify each term.

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Step 4.1

Multiply $- 2 \frac{2}{s^{4}}$ .

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Step 4.1.1

Combine $- 2$ and $\frac{2}{s^{4}}$ .

$\frac{4}{s^{2}} + \frac{- 2 \cdot 2}{s^{4}} + \frac{1}{s^{6}}$

Step 4.1.2

Multiply $- 2$ by $2$ .

$\frac{4}{s^{2}} + \frac{- 4}{s^{4}} + \frac{1}{s^{6}}$

Step 4.2

Move the negative in front of the fraction.

$\frac{4}{s^{2}} - \frac{4}{s^{4}} + \frac{1}{s^{6}}$