Algebra Examples

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

Step 1

Combine fractions.

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Step 1.1

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

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

Step 1.2

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

$(3 m^{3} - \frac{y}{2}) (3 m^{3} - \frac{y}{2})$

Step 2

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

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Step 2.1

Apply the distributive property.

$3 m^{3} (3 m^{3} - \frac{y}{2}) - \frac{y}{2} (3 m^{3} - \frac{y}{2})$

Step 2.2

Apply the distributive property.

$3 m^{3} (3 m^{3}) + 3 m^{3} (- \frac{y}{2}) - \frac{y}{2} (3 m^{3} - \frac{y}{2})$

Step 2.3

Apply the distributive property.

$3 m^{3} (3 m^{3}) + 3 m^{3} (- \frac{y}{2}) - \frac{y}{2} (3 m^{3}) - \frac{y}{2} (- \frac{y}{2})$

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

Rewrite using the commutative property of multiplication.

$3 \cdot 3 m^{3} m^{3} + 3 m^{3} (- \frac{y}{2}) - \frac{y}{2} (3 m^{3}) - \frac{y}{2} (- \frac{y}{2})$

Step 3.1.2

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

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Step 3.1.2.1

Move $m^{3}$ .

$3 \cdot 3 (m^{3} m^{3}) + 3 m^{3} (- \frac{y}{2}) - \frac{y}{2} (3 m^{3}) - \frac{y}{2} (- \frac{y}{2})$

Step 3.1.2.2

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

$3 \cdot 3 m^{3 + 3} + 3 m^{3} (- \frac{y}{2}) - \frac{y}{2} (3 m^{3}) - \frac{y}{2} (- \frac{y}{2})$

Step 3.1.2.3

Add $3$ and $3$ .

$3 \cdot 3 m^{6} + 3 m^{3} (- \frac{y}{2}) - \frac{y}{2} (3 m^{3}) - \frac{y}{2} (- \frac{y}{2})$

Step 3.1.3

Multiply $3$ by $3$ .

$9 m^{6} + 3 m^{3} (- \frac{y}{2}) - \frac{y}{2} (3 m^{3}) - \frac{y}{2} (- \frac{y}{2})$

Step 3.1.4

Multiply $3 m^{3} (- \frac{y}{2})$ .

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Step 3.1.4.1

Multiply $- 1$ by $3$ .

$9 m^{6} - 3 m^{3} \frac{y}{2} - \frac{y}{2} (3 m^{3}) - \frac{y}{2} (- \frac{y}{2})$

Step 3.1.4.2

Combine $- 3$ and $\frac{y}{2}$ .

$9 m^{6} + m^{3} \frac{- 3 y}{2} - \frac{y}{2} (3 m^{3}) - \frac{y}{2} (- \frac{y}{2})$

Step 3.1.4.3

Combine $m^{3}$ and $\frac{- 3 y}{2}$ .

$9 m^{6} + \frac{m^{3} (- 3 y)}{2} - \frac{y}{2} (3 m^{3}) - \frac{y}{2} (- \frac{y}{2})$

Step 3.1.5

Remove unnecessary parentheses.

$9 m^{6} + \frac{m^{3} \cdot - 3 y}{2} - \frac{y}{2} (3 m^{3}) - \frac{y}{2} (- \frac{y}{2})$

Step 3.1.6

Move $- 3$ to the left of $m^{3}$ .

$9 m^{6} + \frac{- 3 \cdot m^{3} y}{2} - \frac{y}{2} (3 m^{3}) - \frac{y}{2} (- \frac{y}{2})$

Step 3.1.7

Move the negative in front of the fraction.

$9 m^{6} - \frac{3 \cdot m^{3} y}{2} - \frac{y}{2} (3 m^{3}) - \frac{y}{2} (- \frac{y}{2})$

Step 3.1.8

Multiply $- \frac{y}{2} (3 m^{3})$ .

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Step 3.1.8.1

Multiply $3$ by $- 1$ .

$9 m^{6} - \frac{3 m^{3} y}{2} - 3 \frac{y}{2} m^{3} - \frac{y}{2} (- \frac{y}{2})$

Step 3.1.8.2

Combine $- 3$ and $\frac{y}{2}$ .

$9 m^{6} - \frac{3 m^{3} y}{2} + \frac{- 3 y}{2} m^{3} - \frac{y}{2} (- \frac{y}{2})$

Step 3.1.8.3

Combine $\frac{- 3 y}{2}$ and $m^{3}$ .

$9 m^{6} - \frac{3 m^{3} y}{2} + \frac{- 3 y m^{3}}{2} - \frac{y}{2} (- \frac{y}{2})$

Step 3.1.9

Move the negative in front of the fraction.

$9 m^{6} - \frac{3 m^{3} y}{2} - \frac{3 y m^{3}}{2} - \frac{y}{2} (- \frac{y}{2})$

Step 3.1.10

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

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Step 3.1.10.1

Multiply $- 1$ by $- 1$ .

$9 m^{6} - \frac{3 m^{3} y}{2} - \frac{3 y m^{3}}{2} + 1 \frac{y}{2} \frac{y}{2}$

Step 3.1.10.2

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

$9 m^{6} - \frac{3 m^{3} y}{2} - \frac{3 y m^{3}}{2} + \frac{y}{2} \cdot \frac{y}{2}$

Step 3.1.10.3

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

$9 m^{6} - \frac{3 m^{3} y}{2} - \frac{3 y m^{3}}{2} + \frac{y \cdot y}{2 \cdot 2}$

Step 3.1.10.4

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

$9 m^{6} - \frac{3 m^{3} y}{2} - \frac{3 y m^{3}}{2} + \frac{y^{1} y}{2 \cdot 2}$

Step 3.1.10.5

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

$9 m^{6} - \frac{3 m^{3} y}{2} - \frac{3 y m^{3}}{2} + \frac{y^{1} y^{1}}{2 \cdot 2}$

Step 3.1.10.6

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

$9 m^{6} - \frac{3 m^{3} y}{2} - \frac{3 y m^{3}}{2} + \frac{y^{1 + 1}}{2 \cdot 2}$

Step 3.1.10.7

Add $1$ and $1$ .

$9 m^{6} - \frac{3 m^{3} y}{2} - \frac{3 y m^{3}}{2} + \frac{y^{2}}{2 \cdot 2}$

Step 3.1.10.8

Multiply $2$ by $2$ .

$9 m^{6} - \frac{3 m^{3} y}{2} - \frac{3 y m^{3}}{2} + \frac{y^{2}}{4}$

Step 3.2

Subtract $\frac{3 y m^{3}}{2}$ from $- \frac{3 m^{3} y}{2}$ .

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Step 3.2.1

Move $y$ .

$9 m^{6} - \frac{3 m^{3} y}{2} - \frac{3 m^{3} y}{2} + \frac{y^{2}}{4}$

Step 3.2.2

Subtract $\frac{3 m^{3} y}{2}$ from $- \frac{3 m^{3} y}{2}$ .

$9 m^{6} - 2 \frac{3 m^{3} y}{2} + \frac{y^{2}}{4}$

Step 4

Simplify each term.

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Step 4.1

Cancel the common factor of $2$ .

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Step 4.1.1

Factor $2$ out of $- 2$ .

$9 m^{6} + 2 (- 1) \frac{3 m^{3} y}{2} + \frac{y^{2}}{4}$

Step 4.1.2

Cancel the common factor.

$9 m^{6} + 2 \cdot - 1 \frac{3 m^{3} y}{2} + \frac{y^{2}}{4}$

Step 4.1.3

Rewrite the expression.

$9 m^{6} - 1 (3 m^{3} y) + \frac{y^{2}}{4}$

Step 4.2

Multiply $3$ by $- 1$ .

$9 m^{6} - 3 m^{3} y + \frac{y^{2}}{4}$