Pre-Algebra Examples

$\frac{4}{5} \cdot (a b^{2}) - \frac{3}{8} \cdot (a b^{2}) + \frac{1}{2} \cdot (a^{2} b) - \frac{5}{4} \cdot (a b^{2})$

Step 1

Simplify each term.

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Step 1.1

Multiply $\frac{4}{5} (a b^{2})$ .

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Step 1.1.1

Combine $a$ and $\frac{4}{5}$ .

$\frac{a \cdot 4}{5} b^{2} - \frac{3}{8} \cdot (a b^{2}) + \frac{1}{2} \cdot (a^{2} b) - \frac{5}{4} \cdot (a b^{2})$

Step 1.1.2

Combine $\frac{a \cdot 4}{5}$ and $b^{2}$ .

$\frac{a \cdot 4 b^{2}}{5} - \frac{3}{8} \cdot (a b^{2}) + \frac{1}{2} \cdot (a^{2} b) - \frac{5}{4} \cdot (a b^{2})$

Step 1.2

Move $4$ to the left of $a$ .

$\frac{4 \cdot a b^{2}}{5} - \frac{3}{8} \cdot (a b^{2}) + \frac{1}{2} \cdot (a^{2} b) - \frac{5}{4} \cdot (a b^{2})$

Step 1.3

Multiply $- \frac{3}{8} (a b^{2})$ .

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Step 1.3.1

Combine $a$ and $\frac{3}{8}$ .

$\frac{4 a b^{2}}{5} - \frac{a \cdot 3}{8} b^{2} + \frac{1}{2} \cdot (a^{2} b) - \frac{5}{4} \cdot (a b^{2})$

Step 1.3.2

Combine $b^{2}$ and $\frac{a \cdot 3}{8}$ .

$\frac{4 a b^{2}}{5} - \frac{b^{2} (a \cdot 3)}{8} + \frac{1}{2} \cdot (a^{2} b) - \frac{5}{4} \cdot (a b^{2})$

Step 1.4

Move $3$ to the left of $b^{2} a$ .

$\frac{4 a b^{2}}{5} - \frac{3 \cdot (b^{2} a)}{8} + \frac{1}{2} \cdot (a^{2} b) - \frac{5}{4} \cdot (a b^{2})$

Step 1.5

Multiply $\frac{1}{2} (a^{2} b)$ .

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Step 1.5.1

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

$\frac{4 a b^{2}}{5} - \frac{3 b^{2} a}{8} + \frac{a^{2}}{2} b - \frac{5}{4} \cdot (a b^{2})$

Step 1.5.2

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

$\frac{4 a b^{2}}{5} - \frac{3 b^{2} a}{8} + \frac{a^{2} b}{2} - \frac{5}{4} \cdot (a b^{2})$

Step 1.6

Multiply $- \frac{5}{4} (a b^{2})$ .

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Step 1.6.1

Combine $a$ and $\frac{5}{4}$ .

$\frac{4 a b^{2}}{5} - \frac{3 b^{2} a}{8} + \frac{a^{2} b}{2} - \frac{a \cdot 5}{4} b^{2}$

Step 1.6.2

Combine $b^{2}$ and $\frac{a \cdot 5}{4}$ .

$\frac{4 a b^{2}}{5} - \frac{3 b^{2} a}{8} + \frac{a^{2} b}{2} - \frac{b^{2} (a \cdot 5)}{4}$

Step 1.7

Move $5$ to the left of $b^{2} a$ .

$\frac{4 a b^{2}}{5} - \frac{3 b^{2} a}{8} + \frac{a^{2} b}{2} - \frac{5 b^{2} a}{4}$

Step 2

Subtract $\frac{3 b^{2} a}{8}$ from $\frac{4 a b^{2}}{5}$ .

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Step 2.1

Move $b^{2}$ .

$\frac{4 a b^{2}}{5} - \frac{3 a b^{2}}{8} + \frac{a^{2} b}{2} - \frac{5 b^{2} a}{4}$

Step 2.2

To write $\frac{4 a b^{2}}{5}$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$\frac{4 a b^{2}}{5} \cdot \frac{8}{8} - \frac{3 a b^{2}}{8} + \frac{a^{2} b}{2} - \frac{5 b^{2} a}{4}$

Step 2.3

To write $- \frac{3 a b^{2}}{8}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{4 a b^{2}}{5} \cdot \frac{8}{8} - \frac{3 a b^{2}}{8} \cdot \frac{5}{5} + \frac{a^{2} b}{2} - \frac{5 b^{2} a}{4}$

Step 2.4

Write each expression with a common denominator of $40$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.4.1

Multiply $\frac{4 a b^{2}}{5}$ by $\frac{8}{8}$ .

$\frac{4 a b^{2} \cdot 8}{5 \cdot 8} - \frac{3 a b^{2}}{8} \cdot \frac{5}{5} + \frac{a^{2} b}{2} - \frac{5 b^{2} a}{4}$

Step 2.4.2

Multiply $5$ by $8$ .

$\frac{4 a b^{2} \cdot 8}{40} - \frac{3 a b^{2}}{8} \cdot \frac{5}{5} + \frac{a^{2} b}{2} - \frac{5 b^{2} a}{4}$

Step 2.4.3

Multiply $\frac{3 a b^{2}}{8}$ by $\frac{5}{5}$ .

$\frac{4 a b^{2} \cdot 8}{40} - \frac{3 a b^{2} \cdot 5}{8 \cdot 5} + \frac{a^{2} b}{2} - \frac{5 b^{2} a}{4}$

Step 2.4.4

Multiply $8$ by $5$ .

$\frac{4 a b^{2} \cdot 8}{40} - \frac{3 a b^{2} \cdot 5}{40} + \frac{a^{2} b}{2} - \frac{5 b^{2} a}{4}$

Step 2.5

Combine the numerators over the common denominator.

$\frac{4 a b^{2} \cdot 8 - 3 a b^{2} \cdot 5}{40} + \frac{a^{2} b}{2} - \frac{5 b^{2} a}{4}$

Step 3

Simplify each term.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Factor $a b^{2}$ out of $4 a b^{2} \cdot 8 - 3 a b^{2} \cdot 5$ .

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Step 3.1.1.1

Factor $a b^{2}$ out of $4 a b^{2} \cdot 8$ .

$\frac{a b^{2} (4 \cdot 8) - 3 a b^{2} \cdot 5}{40} + \frac{a^{2} b}{2} - \frac{5 b^{2} a}{4}$

Step 3.1.1.2

Factor $a b^{2}$ out of $- 3 a b^{2} \cdot 5$ .

$\frac{a b^{2} (4 \cdot 8) + a b^{2} (- 3 \cdot 5)}{40} + \frac{a^{2} b}{2} - \frac{5 b^{2} a}{4}$

Step 3.1.1.3

Factor $a b^{2}$ out of $a b^{2} (4 \cdot 8) + a b^{2} (- 3 \cdot 5)$ .

$\frac{a b^{2} (4 \cdot 8 - 3 \cdot 5)}{40} + \frac{a^{2} b}{2} - \frac{5 b^{2} a}{4}$

Step 3.1.2

Combine exponents.

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Step 3.1.2.1

Multiply $4$ by $8$ .

$\frac{a b^{2} (32 - 3 \cdot 5)}{40} + \frac{a^{2} b}{2} - \frac{5 b^{2} a}{4}$

Step 3.1.2.2

Multiply $- 3$ by $5$ .

$\frac{a b^{2} (32 - 15)}{40} + \frac{a^{2} b}{2} - \frac{5 b^{2} a}{4}$

Step 3.1.3

Subtract $15$ from $32$ .

$\frac{a b^{2} \cdot 17}{40} + \frac{a^{2} b}{2} - \frac{5 b^{2} a}{4}$

Step 3.2

Move $17$ to the left of $a b^{2}$ .

$\frac{17 a b^{2}}{40} + \frac{a^{2} b}{2} - \frac{5 b^{2} a}{4}$

Step 4

Subtract $\frac{5 b^{2} a}{4}$ from $\frac{17 a b^{2}}{40}$ .

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Step 4.1

Move $b^{2}$ .

$\frac{a^{2} b}{2} + \frac{17 a b^{2}}{40} - \frac{5 a b^{2}}{4}$

Step 4.2

To write $- \frac{5 a b^{2}}{4}$ as a fraction with a common denominator, multiply by $\frac{10}{10}$ .

$\frac{a^{2} b}{2} + \frac{17 a b^{2}}{40} - \frac{5 a b^{2}}{4} \cdot \frac{10}{10}$

Step 4.3

Write each expression with a common denominator of $40$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.3.1

Multiply $\frac{5 a b^{2}}{4}$ by $\frac{10}{10}$ .

$\frac{a^{2} b}{2} + \frac{17 a b^{2}}{40} - \frac{5 a b^{2} \cdot 10}{4 \cdot 10}$

Step 4.3.2

Multiply $4$ by $10$ .

$\frac{a^{2} b}{2} + \frac{17 a b^{2}}{40} - \frac{5 a b^{2} \cdot 10}{40}$

Step 4.4

Combine the numerators over the common denominator.

$\frac{a^{2} b}{2} + \frac{17 a b^{2} - 5 a b^{2} \cdot 10}{40}$

Step 5

Simplify each term.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Factor $a b^{2}$ out of $17 a b^{2} - 5 a b^{2} \cdot 10$ .

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Step 5.1.1.1

Factor $a b^{2}$ out of $17 a b^{2}$ .

$\frac{a^{2} b}{2} + \frac{a b^{2} (17) - 5 a b^{2} \cdot 10}{40}$

Step 5.1.1.2

Factor $a b^{2}$ out of $- 5 a b^{2} \cdot 10$ .

$\frac{a^{2} b}{2} + \frac{a b^{2} (17) + a b^{2} (- 5 \cdot 10)}{40}$

Step 5.1.1.3

Factor $a b^{2}$ out of $a b^{2} (17) + a b^{2} (- 5 \cdot 10)$ .

$\frac{a^{2} b}{2} + \frac{a b^{2} (17 - 5 \cdot 10)}{40}$

Step 5.1.2

Multiply $- 5$ by $10$ .

$\frac{a^{2} b}{2} + \frac{a b^{2} (17 - 50)}{40}$

Step 5.1.3

Subtract $50$ from $17$ .

$\frac{a^{2} b}{2} + \frac{a b^{2} \cdot - 33}{40}$

Step 5.2

Move $- 33$ to the left of $a b^{2}$ .

$\frac{a^{2} b}{2} + \frac{- 33 \cdot (a b^{2})}{40}$

Step 5.3

Move the negative in front of the fraction.

$\frac{a^{2} b}{2} - \frac{33 a b^{2}}{40}$