Basic Math Examples

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

Step 1

Simplify terms.

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Step 1.1

Simplify each term.

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Step 1.1.1

Combine $a$ and $\frac{1}{2}$ .

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

Step 1.1.2

Combine $b$ and $\frac{1}{3}$ .

${(- \frac{a}{2} - \frac{b}{3})}^{2}$

Step 1.2

Rewrite ${(- \frac{a}{2} - \frac{b}{3})}^{2}$ as $(- \frac{a}{2} - \frac{b}{3}) (- \frac{a}{2} - \frac{b}{3})$ .

$(- \frac{a}{2} - \frac{b}{3}) (- \frac{a}{2} - \frac{b}{3})$

Step 2

Expand $(- \frac{a}{2} - \frac{b}{3}) (- \frac{a}{2} - \frac{b}{3})$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$- \frac{a}{2} (- \frac{a}{2} - \frac{b}{3}) - \frac{b}{3} (- \frac{a}{2} - \frac{b}{3})$

Step 2.2

Apply the distributive property.

$- \frac{a}{2} (- \frac{a}{2}) - \frac{a}{2} (- \frac{b}{3}) - \frac{b}{3} (- \frac{a}{2} - \frac{b}{3})$

Step 2.3

Apply the distributive property.

$- \frac{a}{2} (- \frac{a}{2}) - \frac{a}{2} (- \frac{b}{3}) - \frac{b}{3} (- \frac{a}{2}) - \frac{b}{3} (- \frac{b}{3})$

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 $- \frac{a}{2} (- \frac{a}{2})$ .

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Step 3.1.1.1

Multiply $- 1$ by $- 1$ .

$1 \frac{a}{2} \frac{a}{2} - \frac{a}{2} (- \frac{b}{3}) - \frac{b}{3} (- \frac{a}{2}) - \frac{b}{3} (- \frac{b}{3})$

Step 3.1.1.2

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

$\frac{a}{2} \cdot \frac{a}{2} - \frac{a}{2} (- \frac{b}{3}) - \frac{b}{3} (- \frac{a}{2}) - \frac{b}{3} (- \frac{b}{3})$

Step 3.1.1.3

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

$\frac{a \cdot a}{2 \cdot 2} - \frac{a}{2} (- \frac{b}{3}) - \frac{b}{3} (- \frac{a}{2}) - \frac{b}{3} (- \frac{b}{3})$

Step 3.1.1.4

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

$\frac{a^{1} a}{2 \cdot 2} - \frac{a}{2} (- \frac{b}{3}) - \frac{b}{3} (- \frac{a}{2}) - \frac{b}{3} (- \frac{b}{3})$

Step 3.1.1.5

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

$\frac{a^{1} a^{1}}{2 \cdot 2} - \frac{a}{2} (- \frac{b}{3}) - \frac{b}{3} (- \frac{a}{2}) - \frac{b}{3} (- \frac{b}{3})$

Step 3.1.1.6

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

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

Step 3.1.1.7

Add $1$ and $1$ .

$\frac{a^{2}}{2 \cdot 2} - \frac{a}{2} (- \frac{b}{3}) - \frac{b}{3} (- \frac{a}{2}) - \frac{b}{3} (- \frac{b}{3})$

Step 3.1.1.8

Multiply $2$ by $2$ .

$\frac{a^{2}}{4} - \frac{a}{2} (- \frac{b}{3}) - \frac{b}{3} (- \frac{a}{2}) - \frac{b}{3} (- \frac{b}{3})$

Step 3.1.2

Multiply $- \frac{a}{2} (- \frac{b}{3})$ .

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Step 3.1.2.1

Multiply $- 1$ by $- 1$ .

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

Step 3.1.2.2

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

$\frac{a^{2}}{4} + \frac{a}{2} \cdot \frac{b}{3} - \frac{b}{3} (- \frac{a}{2}) - \frac{b}{3} (- \frac{b}{3})$

Step 3.1.2.3

Multiply $\frac{a}{2}$ by $\frac{b}{3}$ .

$\frac{a^{2}}{4} + \frac{a b}{2 \cdot 3} - \frac{b}{3} (- \frac{a}{2}) - \frac{b}{3} (- \frac{b}{3})$

Step 3.1.2.4

Multiply $2$ by $3$ .

$\frac{a^{2}}{4} + \frac{a b}{6} - \frac{b}{3} (- \frac{a}{2}) - \frac{b}{3} (- \frac{b}{3})$

Step 3.1.3

Multiply $- \frac{b}{3} (- \frac{a}{2})$ .

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Step 3.1.3.1

Multiply $- 1$ by $- 1$ .

$\frac{a^{2}}{4} + \frac{a b}{6} + 1 \frac{b}{3} \frac{a}{2} - \frac{b}{3} (- \frac{b}{3})$

Step 3.1.3.2

Multiply $\frac{b}{3}$ by $1$ .

$\frac{a^{2}}{4} + \frac{a b}{6} + \frac{b}{3} \cdot \frac{a}{2} - \frac{b}{3} (- \frac{b}{3})$

Step 3.1.3.3

Multiply $\frac{b}{3}$ by $\frac{a}{2}$ .

$\frac{a^{2}}{4} + \frac{a b}{6} + \frac{b a}{3 \cdot 2} - \frac{b}{3} (- \frac{b}{3})$

Step 3.1.3.4

Multiply $3$ by $2$ .

$\frac{a^{2}}{4} + \frac{a b}{6} + \frac{b a}{6} - \frac{b}{3} (- \frac{b}{3})$

Step 3.1.4

Multiply $- \frac{b}{3} (- \frac{b}{3})$ .

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Step 3.1.4.1

Multiply $- 1$ by $- 1$ .

$\frac{a^{2}}{4} + \frac{a b}{6} + \frac{b a}{6} + 1 \frac{b}{3} \frac{b}{3}$

Step 3.1.4.2

Multiply $\frac{b}{3}$ by $1$ .

$\frac{a^{2}}{4} + \frac{a b}{6} + \frac{b a}{6} + \frac{b}{3} \cdot \frac{b}{3}$

Step 3.1.4.3

Multiply $\frac{b}{3}$ by $\frac{b}{3}$ .

$\frac{a^{2}}{4} + \frac{a b}{6} + \frac{b a}{6} + \frac{b \cdot b}{3 \cdot 3}$

Step 3.1.4.4

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

$\frac{a^{2}}{4} + \frac{a b}{6} + \frac{b a}{6} + \frac{b^{1} b}{3 \cdot 3}$

Step 3.1.4.5

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

$\frac{a^{2}}{4} + \frac{a b}{6} + \frac{b a}{6} + \frac{b^{1} b^{1}}{3 \cdot 3}$

Step 3.1.4.6

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

$\frac{a^{2}}{4} + \frac{a b}{6} + \frac{b a}{6} + \frac{b^{1 + 1}}{3 \cdot 3}$

Step 3.1.4.7

Add $1$ and $1$ .

$\frac{a^{2}}{4} + \frac{a b}{6} + \frac{b a}{6} + \frac{b^{2}}{3 \cdot 3}$

Step 3.1.4.8

Multiply $3$ by $3$ .

$\frac{a^{2}}{4} + \frac{a b}{6} + \frac{b a}{6} + \frac{b^{2}}{9}$

Step 3.2

Add $\frac{a b}{6}$ and $\frac{b a}{6}$ .

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Step 3.2.1

Reorder $b$ and $a$ .

$\frac{a^{2}}{4} + \frac{a b}{6} + \frac{a b}{6} + \frac{b^{2}}{9}$

Step 3.2.2

Add $\frac{a b}{6}$ and $\frac{a b}{6}$ .

$\frac{a^{2}}{4} + 2 \frac{a b}{6} + \frac{b^{2}}{9}$

Step 4

Cancel the common factor of $2$ .

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Step 4.1

Factor $2$ out of $6$ .

$\frac{a^{2}}{4} + 2 \frac{a b}{2 (3)} + \frac{b^{2}}{9}$

Step 4.2

Cancel the common factor.

$\frac{a^{2}}{4} + 2 \frac{a b}{2 \cdot 3} + \frac{b^{2}}{9}$

Step 4.3

Rewrite the expression.

$\frac{a^{2}}{4} + \frac{a b}{3} + \frac{b^{2}}{9}$