Algebra Examples

$\frac{1}{4} a^{3} b - \frac{3}{5} a^{3} b + \frac{1}{6} a^{3} b$

Step 1

Simplify each term.

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Step 1.1

Multiply $\frac{1}{4} (a^{3} b)$ .

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Step 1.1.1

Combine $a^{3}$ and $\frac{1}{4}$ .

$\frac{a^{3}}{4} b - \frac{3}{5} \cdot (a^{3} b) + \frac{1}{6} \cdot (a^{3} b)$

Step 1.1.2

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

$\frac{a^{3} b}{4} - \frac{3}{5} \cdot (a^{3} b) + \frac{1}{6} \cdot (a^{3} b)$

Step 1.2

Multiply $- \frac{3}{5} (a^{3} b)$ .

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Step 1.2.1

Combine $a^{3}$ and $\frac{3}{5}$ .

$\frac{a^{3} b}{4} - \frac{a^{3} \cdot 3}{5} b + \frac{1}{6} \cdot (a^{3} b)$

Step 1.2.2

Combine $b$ and $\frac{a^{3} \cdot 3}{5}$ .

$\frac{a^{3} b}{4} - \frac{b (a^{3} \cdot 3)}{5} + \frac{1}{6} \cdot (a^{3} b)$

Step 1.3

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

$\frac{a^{3} b}{4} - \frac{3 \cdot (b a^{3})}{5} + \frac{1}{6} \cdot (a^{3} b)$

Step 1.4

Multiply $\frac{1}{6} (a^{3} b)$ .

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Step 1.4.1

Combine $a^{3}$ and $\frac{1}{6}$ .

$\frac{a^{3} b}{4} - \frac{3 b a^{3}}{5} + \frac{a^{3}}{6} b$

Step 1.4.2

Combine $\frac{a^{3}}{6}$ and $b$ .

$\frac{a^{3} b}{4} - \frac{3 b a^{3}}{5} + \frac{a^{3} b}{6}$

Step 2

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

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Step 2.1

Move $b$ .

$\frac{a^{3} b}{4} - \frac{3 a^{3} b}{5} + \frac{a^{3} b}{6}$

Step 2.2

To write $\frac{a^{3} b}{4}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{a^{3} b}{4} \cdot \frac{5}{5} - \frac{3 a^{3} b}{5} + \frac{a^{3} b}{6}$

Step 2.3

To write $- \frac{3 a^{3} b}{5}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{a^{3} b}{4} \cdot \frac{5}{5} - \frac{3 a^{3} b}{5} \cdot \frac{4}{4} + \frac{a^{3} b}{6}$

Step 2.4

Write each expression with a common denominator of $20$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.4.1

Multiply $\frac{a^{3} b}{4}$ by $\frac{5}{5}$ .

$\frac{a^{3} b \cdot 5}{4 \cdot 5} - \frac{3 a^{3} b}{5} \cdot \frac{4}{4} + \frac{a^{3} b}{6}$

Step 2.4.2

Multiply $4$ by $5$ .

$\frac{a^{3} b \cdot 5}{20} - \frac{3 a^{3} b}{5} \cdot \frac{4}{4} + \frac{a^{3} b}{6}$

Step 2.4.3

Multiply $\frac{3 a^{3} b}{5}$ by $\frac{4}{4}$ .

$\frac{a^{3} b \cdot 5}{20} - \frac{3 a^{3} b \cdot 4}{5 \cdot 4} + \frac{a^{3} b}{6}$

Step 2.4.4

Multiply $5$ by $4$ .

$\frac{a^{3} b \cdot 5}{20} - \frac{3 a^{3} b \cdot 4}{20} + \frac{a^{3} b}{6}$

Step 2.5

Combine the numerators over the common denominator.

$\frac{a^{3} b \cdot 5 - 3 a^{3} b \cdot 4}{20} + \frac{a^{3} b}{6}$

Step 3

Simplify each term.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

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

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Step 3.1.1.1

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

$\frac{a^{3} b (5) - 3 a^{3} b \cdot 4}{20} + \frac{a^{3} b}{6}$

Step 3.1.1.2

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

$\frac{a^{3} b (5) + a^{3} b (- 3 \cdot 4)}{20} + \frac{a^{3} b}{6}$

Step 3.1.1.3

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

$\frac{a^{3} b (5 - 3 \cdot 4)}{20} + \frac{a^{3} b}{6}$

Step 3.1.2

Multiply $- 3$ by $4$ .

$\frac{a^{3} b (5 - 12)}{20} + \frac{a^{3} b}{6}$

Step 3.1.3

Subtract $12$ from $5$ .

$\frac{a^{3} b \cdot - 7}{20} + \frac{a^{3} b}{6}$

Step 3.2

Move $- 7$ to the left of $a^{3} b$ .

$\frac{- 7 \cdot (a^{3} b)}{20} + \frac{a^{3} b}{6}$

Step 3.3

Move the negative in front of the fraction.

$- \frac{7 a^{3} b}{20} + \frac{a^{3} b}{6}$

Step 4

To write $- \frac{7 a^{3} b}{20}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- \frac{7 a^{3} b}{20} \cdot \frac{3}{3} + \frac{a^{3} b}{6}$

Step 5

To write $\frac{a^{3} b}{6}$ as a fraction with a common denominator, multiply by $\frac{10}{10}$ .

$- \frac{7 a^{3} b}{20} \cdot \frac{3}{3} + \frac{a^{3} b}{6} \cdot \frac{10}{10}$

Step 6

Write each expression with a common denominator of $60$ , by multiplying each by an appropriate factor of $1$ .

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Step 6.1

Multiply $\frac{7 a^{3} b}{20}$ by $\frac{3}{3}$ .

$- \frac{7 a^{3} b \cdot 3}{20 \cdot 3} + \frac{a^{3} b}{6} \cdot \frac{10}{10}$

Step 6.2

Multiply $20$ by $3$ .

$- \frac{7 a^{3} b \cdot 3}{60} + \frac{a^{3} b}{6} \cdot \frac{10}{10}$

Step 6.3

Multiply $\frac{a^{3} b}{6}$ by $\frac{10}{10}$ .

$- \frac{7 a^{3} b \cdot 3}{60} + \frac{a^{3} b \cdot 10}{6 \cdot 10}$

Step 6.4

Multiply $6$ by $10$ .

$- \frac{7 a^{3} b \cdot 3}{60} + \frac{a^{3} b \cdot 10}{60}$

Step 7

Combine the numerators over the common denominator.

$\frac{- 7 a^{3} b \cdot 3 + a^{3} b \cdot 10}{60}$

Step 8

Simplify the numerator.

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Step 8.1

Factor $a^{3} b$ out of $- 7 a^{3} b \cdot 3 + a^{3} b \cdot 10$ .

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Step 8.1.1

Factor $a^{3} b$ out of $- 7 a^{3} b \cdot 3$ .

$\frac{a^{3} b (- 7 \cdot 3) + a^{3} b \cdot 10}{60}$

Step 8.1.2

Factor $a^{3} b$ out of $a^{3} b \cdot 10$ .

$\frac{a^{3} b (- 7 \cdot 3) + a^{3} b (10)}{60}$

Step 8.1.3

Factor $a^{3} b$ out of $a^{3} b (- 7 \cdot 3) + a^{3} b (10)$ .

$\frac{a^{3} b (- 7 \cdot 3 + 10)}{60}$

Step 8.2

Multiply $- 7$ by $3$ .

$\frac{a^{3} b (- 21 + 10)}{60}$

Step 8.3

Add $- 21$ and $10$ .

$\frac{a^{3} b \cdot - 11}{60}$

Step 9

Simplify the expression.

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Step 9.1

Move $- 11$ to the left of $a^{3} b$ .

$\frac{- 11 \cdot (a^{3} b)}{60}$

Step 9.2

Move the negative in front of the fraction.

$- \frac{11 a^{3} b}{60}$