Basic Math Examples

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

Step 1

Simplify terms.

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Step 1.1

Simplify each term.

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Step 1.1.1

Combine $\frac{2}{3}$ and $a$ .

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

Step 1.1.2

Combine $b$ and $\frac{3}{4}$ .

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

Step 1.1.3

Move $3$ to the left of $b$ .

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

Step 1.2

Simplify each term.

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Step 1.2.1

Combine $\frac{2}{3}$ and $a$ .

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

Step 1.2.2

Combine $b$ and $\frac{3}{4}$ .

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

Step 1.2.3

Move $3$ to the left of $b$ .

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

Step 2

Expand $(\frac{2 a}{3} - \frac{3 b}{4}) (\frac{2 a}{3} - \frac{3 b}{4})$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

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

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

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

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 $\frac{2 a}{3} \cdot \frac{2 a}{3}$ .

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Step 3.1.1.1

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

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

Step 3.1.1.2

Multiply $2$ by $2$ .

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

Step 3.1.1.3

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

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

Step 3.1.1.4

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

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

Step 3.1.1.5

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

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

Step 3.1.1.6

Add $1$ and $1$ .

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

Step 3.1.1.7

Multiply $3$ by $3$ .

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

Step 3.1.2

Rewrite using the commutative property of multiplication.

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

Step 3.1.3

Cancel the common factor of $2$ .

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Step 3.1.3.1

Move the leading negative in $- \frac{2 a}{3}$ into the numerator.

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

Step 3.1.3.2

Factor $2$ out of $- 2 a$ .

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

Step 3.1.3.3

Factor $2$ out of $4$ .

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

Step 3.1.3.4

Cancel the common factor.

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

Step 3.1.3.5

Rewrite the expression.

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

Step 3.1.4

Cancel the common factor of $3$ .

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Step 3.1.4.1

Factor $3$ out of $3 b$ .

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

Step 3.1.4.2

Cancel the common factor.

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

Step 3.1.4.3

Rewrite the expression.

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

Step 3.1.5

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

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

Step 3.1.6

Cancel the common factor of $3$ .

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Step 3.1.6.1

Move the leading negative in $- \frac{3 b}{4}$ into the numerator.

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

Step 3.1.6.2

Factor $3$ out of $- 3 b$ .

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

Step 3.1.6.3

Cancel the common factor.

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

Step 3.1.6.4

Rewrite the expression.

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

Step 3.1.7

Cancel the common factor of $2$ .

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Step 3.1.7.1

Factor $2$ out of $4$ .

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

Step 3.1.7.2

Factor $2$ out of $2 a$ .

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

Step 3.1.7.3

Cancel the common factor.

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

Step 3.1.7.4

Rewrite the expression.

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

Step 3.1.8

Combine $\frac{- b}{2}$ and $a$ .

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

Step 3.1.9

Move the negative in front of the fraction.

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

Step 3.1.10

Multiply $- \frac{3 b}{4} (- \frac{3 b}{4})$ .

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Step 3.1.10.1

Multiply $- 1$ by $- 1$ .

$\frac{4 a^{2}}{9} - \frac{b a}{2} - \frac{b a}{2} + 1 \frac{3 b}{4} \frac{3 b}{4}$

Step 3.1.10.2

Multiply $\frac{3 b}{4}$ by $1$ .

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

Step 3.1.10.3

Multiply $\frac{3 b}{4}$ by $\frac{3 b}{4}$ .

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

Step 3.1.10.4

Multiply $3$ by $3$ .

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

Step 3.1.10.5

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

$\frac{4 a^{2}}{9} - \frac{b a}{2} - \frac{b a}{2} + \frac{9 (b^{1} b)}{4 \cdot 4}$

Step 3.1.10.6

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

$\frac{4 a^{2}}{9} - \frac{b a}{2} - \frac{b a}{2} + \frac{9 (b^{1} b^{1})}{4 \cdot 4}$

Step 3.1.10.7

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

$\frac{4 a^{2}}{9} - \frac{b a}{2} - \frac{b a}{2} + \frac{9 b^{1 + 1}}{4 \cdot 4}$

Step 3.1.10.8

Add $1$ and $1$ .

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

Step 3.1.10.9

Multiply $4$ by $4$ .

$\frac{4 a^{2}}{9} - \frac{b a}{2} - \frac{b a}{2} + \frac{9 b^{2}}{16}$

Step 3.2

Subtract $\frac{b a}{2}$ from $- \frac{b a}{2}$ .

$\frac{4 a^{2}}{9} - 2 \frac{b a}{2} + \frac{9 b^{2}}{16}$

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 $- 2$ .

$\frac{4 a^{2}}{9} + 2 (- 1) \frac{b a}{2} + \frac{9 b^{2}}{16}$

Step 4.1.2

Cancel the common factor.

$\frac{4 a^{2}}{9} + 2 \cdot - 1 \frac{b a}{2} + \frac{9 b^{2}}{16}$

Step 4.1.3

Rewrite the expression.

$\frac{4 a^{2}}{9} - 1 (b a) + \frac{9 b^{2}}{16}$

Step 4.2

Rewrite $- 1 (b a)$ as $- (b a)$ .

$\frac{4 a^{2}}{9} - b a + \frac{9 b^{2}}{16}$

Step 5

Move $b$ .

$\frac{4 a^{2}}{9} - a b + \frac{9 b^{2}}{16}$