Algebra Examples

$2 a^{2} b - \frac{5}{2} a b^{2} + \frac{7}{12} a b^{2} + \frac{5}{6} a^{2} b$

Step 1

Simplify each term.

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Step 1.1

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

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Step 1.1.1

Combine $a$ and $\frac{5}{2}$ .

$2 a^{2} b - \frac{a \cdot 5}{2} b^{2} + \frac{7}{12} \cdot (a b^{2}) + \frac{5}{6} \cdot (a^{2} b)$

Step 1.1.2

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

$2 a^{2} b - \frac{b^{2} (a \cdot 5)}{2} + \frac{7}{12} \cdot (a b^{2}) + \frac{5}{6} \cdot (a^{2} b)$

Step 1.2

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

$2 a^{2} b - \frac{5 \cdot (b^{2} a)}{2} + \frac{7}{12} \cdot (a b^{2}) + \frac{5}{6} \cdot (a^{2} b)$

Step 1.3

Multiply $\frac{7}{12} (a b^{2})$ .

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Step 1.3.1

Combine $a$ and $\frac{7}{12}$ .

$2 a^{2} b - \frac{5 b^{2} a}{2} + \frac{a \cdot 7}{12} b^{2} + \frac{5}{6} \cdot (a^{2} b)$

Step 1.3.2

Combine $\frac{a \cdot 7}{12}$ and $b^{2}$ .

$2 a^{2} b - \frac{5 b^{2} a}{2} + \frac{a \cdot 7 b^{2}}{12} + \frac{5}{6} \cdot (a^{2} b)$

Step 1.4

Move $7$ to the left of $a$ .

$2 a^{2} b - \frac{5 b^{2} a}{2} + \frac{7 \cdot a b^{2}}{12} + \frac{5}{6} \cdot (a^{2} b)$

Step 1.5

Multiply $\frac{5}{6} (a^{2} b)$ .

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Step 1.5.1

Combine $a^{2}$ and $\frac{5}{6}$ .

$2 a^{2} b - \frac{5 b^{2} a}{2} + \frac{7 a b^{2}}{12} + \frac{a^{2} \cdot 5}{6} b$

Step 1.5.2

Combine $\frac{a^{2} \cdot 5}{6}$ and $b$ .

$2 a^{2} b - \frac{5 b^{2} a}{2} + \frac{7 a b^{2}}{12} + \frac{a^{2} \cdot 5 b}{6}$

Step 1.6

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

$2 a^{2} b - \frac{5 b^{2} a}{2} + \frac{7 a b^{2}}{12} + \frac{5 a^{2} b}{6}$

Step 2

To write $2 a^{2} b$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$- \frac{5 b^{2} a}{2} + \frac{7 a b^{2}}{12} + 2 a^{2} b \cdot \frac{6}{6} + \frac{5 a^{2} b}{6}$

Step 3

Simplify terms.

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Step 3.1

Combine $2 a^{2} b$ and $\frac{6}{6}$ .

$- \frac{5 b^{2} a}{2} + \frac{7 a b^{2}}{12} + \frac{2 a^{2} b \cdot 6}{6} + \frac{5 a^{2} b}{6}$

Step 3.2

Combine the numerators over the common denominator.

$- \frac{5 b^{2} a}{2} + \frac{7 a b^{2}}{12} + \frac{2 a^{2} b \cdot 6 + 5 a^{2} b}{6}$

Step 4

Simplify each term.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Factor $a^{2} b$ out of $2 a^{2} b \cdot 6 + 5 a^{2} b$ .

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Step 4.1.1.1

Factor $a^{2} b$ out of $2 a^{2} b \cdot 6$ .

$- \frac{5 b^{2} a}{2} + \frac{7 a b^{2}}{12} + \frac{a^{2} b (2 \cdot 6) + 5 a^{2} b}{6}$

Step 4.1.1.2

Factor $a^{2} b$ out of $5 a^{2} b$ .

$- \frac{5 b^{2} a}{2} + \frac{7 a b^{2}}{12} + \frac{a^{2} b (2 \cdot 6) + a^{2} b (5)}{6}$

Step 4.1.1.3

Factor $a^{2} b$ out of $a^{2} b (2 \cdot 6) + a^{2} b (5)$ .

$- \frac{5 b^{2} a}{2} + \frac{7 a b^{2}}{12} + \frac{a^{2} b (2 \cdot 6 + 5)}{6}$

Step 4.1.2

Multiply $2$ by $6$ .

$- \frac{5 b^{2} a}{2} + \frac{7 a b^{2}}{12} + \frac{a^{2} b (12 + 5)}{6}$

Step 4.1.3

Add $12$ and $5$ .

$- \frac{5 b^{2} a}{2} + \frac{7 a b^{2}}{12} + \frac{a^{2} b \cdot 17}{6}$

Step 4.2

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

$- \frac{5 b^{2} a}{2} + \frac{7 a b^{2}}{12} + \frac{17 a^{2} b}{6}$

Step 5

Add $- \frac{5 b^{2} a}{2}$ and $\frac{7 a b^{2}}{12}$ .

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Step 5.1

Move $b^{2}$ .

$- \frac{5 a b^{2}}{2} + \frac{7 a b^{2}}{12} + \frac{17 a^{2} b}{6}$

Step 5.2

To write $- \frac{5 a b^{2}}{2}$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$- \frac{5 a b^{2}}{2} \cdot \frac{6}{6} + \frac{7 a b^{2}}{12} + \frac{17 a^{2} b}{6}$

Step 5.3

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.3.1

Multiply $\frac{5 a b^{2}}{2}$ by $\frac{6}{6}$ .

$- \frac{5 a b^{2} \cdot 6}{2 \cdot 6} + \frac{7 a b^{2}}{12} + \frac{17 a^{2} b}{6}$

Step 5.3.2

Multiply $2$ by $6$ .

$- \frac{5 a b^{2} \cdot 6}{12} + \frac{7 a b^{2}}{12} + \frac{17 a^{2} b}{6}$

Step 5.4

Combine the numerators over the common denominator.

$\frac{- 5 a b^{2} \cdot 6 + 7 a b^{2}}{12} + \frac{17 a^{2} b}{6}$

Step 6

Simplify each term.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Factor $a b^{2}$ out of $- 5 a b^{2} \cdot 6 + 7 a b^{2}$ .

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Step 6.1.1.1

Factor $a b^{2}$ out of $- 5 a b^{2} \cdot 6$ .

$\frac{a b^{2} (- 5 \cdot 6) + 7 a b^{2}}{12} + \frac{17 a^{2} b}{6}$

Step 6.1.1.2

Factor $a b^{2}$ out of $7 a b^{2}$ .

$\frac{a b^{2} (- 5 \cdot 6) + a b^{2} (7)}{12} + \frac{17 a^{2} b}{6}$

Step 6.1.1.3

Factor $a b^{2}$ out of $a b^{2} (- 5 \cdot 6) + a b^{2} (7)$ .

$\frac{a b^{2} (- 5 \cdot 6 + 7)}{12} + \frac{17 a^{2} b}{6}$

Step 6.1.2

Multiply $- 5$ by $6$ .

$\frac{a b^{2} (- 30 + 7)}{12} + \frac{17 a^{2} b}{6}$

Step 6.1.3

Add $- 30$ and $7$ .

$\frac{a b^{2} \cdot - 23}{12} + \frac{17 a^{2} b}{6}$

Step 6.2

Move $- 23$ to the left of $a b^{2}$ .

$\frac{- 23 \cdot (a b^{2})}{12} + \frac{17 a^{2} b}{6}$

Step 6.3

Move the negative in front of the fraction.

$- \frac{23 a b^{2}}{12} + \frac{17 a^{2} b}{6}$