Algebra Examples

$\frac{1}{5} p q - \frac{3}{10} p q^{3} - \frac{3}{5} p q^{3} - \frac{7}{10} p q + 3 p q$

Step 1

Simplify each term.

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Step 1.1

Multiply $\frac{1}{5} (p q)$ .

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Step 1.1.1

Combine $p$ and $\frac{1}{5}$ .

$\frac{p}{5} q - \frac{3}{10} \cdot (p q^{3}) - \frac{3}{5} \cdot (p q^{3}) - \frac{7}{10} \cdot (p q) + 3 p q$

Step 1.1.2

Combine $\frac{p}{5}$ and $q$ .

$\frac{p q}{5} - \frac{3}{10} \cdot (p q^{3}) - \frac{3}{5} \cdot (p q^{3}) - \frac{7}{10} \cdot (p q) + 3 p q$

Step 1.2

Multiply $- \frac{3}{10} (p q^{3})$ .

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Step 1.2.1

Combine $p$ and $\frac{3}{10}$ .

$\frac{p q}{5} - \frac{p \cdot 3}{10} q^{3} - \frac{3}{5} \cdot (p q^{3}) - \frac{7}{10} \cdot (p q) + 3 p q$

Step 1.2.2

Combine $q^{3}$ and $\frac{p \cdot 3}{10}$ .

$\frac{p q}{5} - \frac{q^{3} (p \cdot 3)}{10} - \frac{3}{5} \cdot (p q^{3}) - \frac{7}{10} \cdot (p q) + 3 p q$

Step 1.3

Move $3$ to the left of $q^{3} p$ .

$\frac{p q}{5} - \frac{3 \cdot (q^{3} p)}{10} - \frac{3}{5} \cdot (p q^{3}) - \frac{7}{10} \cdot (p q) + 3 p q$

Step 1.4

Multiply $- \frac{3}{5} (p q^{3})$ .

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Step 1.4.1

Combine $p$ and $\frac{3}{5}$ .

$\frac{p q}{5} - \frac{3 q^{3} p}{10} - \frac{p \cdot 3}{5} q^{3} - \frac{7}{10} \cdot (p q) + 3 p q$

Step 1.4.2

Combine $q^{3}$ and $\frac{p \cdot 3}{5}$ .

$\frac{p q}{5} - \frac{3 q^{3} p}{10} - \frac{q^{3} (p \cdot 3)}{5} - \frac{7}{10} \cdot (p q) + 3 p q$

Step 1.5

Move $3$ to the left of $q^{3} p$ .

$\frac{p q}{5} - \frac{3 q^{3} p}{10} - \frac{3 \cdot (q^{3} p)}{5} - \frac{7}{10} \cdot (p q) + 3 p q$

Step 1.6

Multiply $- \frac{7}{10} (p q)$ .

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Step 1.6.1

Combine $p$ and $\frac{7}{10}$ .

$\frac{p q}{5} - \frac{3 q^{3} p}{10} - \frac{3 q^{3} p}{5} - \frac{p \cdot 7}{10} q + 3 p q$

Step 1.6.2

Combine $q$ and $\frac{p \cdot 7}{10}$ .

$\frac{p q}{5} - \frac{3 q^{3} p}{10} - \frac{3 q^{3} p}{5} - \frac{q (p \cdot 7)}{10} + 3 p q$

Step 1.7

Move $7$ to the left of $q p$ .

$\frac{p q}{5} - \frac{3 q^{3} p}{10} - \frac{3 q^{3} p}{5} - \frac{7 q p}{10} + 3 p q$

Step 2

Subtract $\frac{7 q p}{10}$ from $\frac{p q}{5}$ .

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Step 2.1

Move $q$ .

$- \frac{3 q^{3} p}{10} - \frac{3 q^{3} p}{5} + \frac{p q}{5} - \frac{7 p q}{10} + 3 p q$

Step 2.2

To write $\frac{p q}{5}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \frac{3 q^{3} p}{10} - \frac{3 q^{3} p}{5} + \frac{p q}{5} \cdot \frac{2}{2} - \frac{7 p q}{10} + 3 p q$

Step 2.3

Write each expression with a common denominator of $10$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.3.1

Multiply $\frac{p q}{5}$ by $\frac{2}{2}$ .

$- \frac{3 q^{3} p}{10} - \frac{3 q^{3} p}{5} + \frac{p q \cdot 2}{5 \cdot 2} - \frac{7 p q}{10} + 3 p q$

Step 2.3.2

Multiply $5$ by $2$ .

$- \frac{3 q^{3} p}{10} - \frac{3 q^{3} p}{5} + \frac{p q \cdot 2}{10} - \frac{7 p q}{10} + 3 p q$

Step 2.4

Combine the numerators over the common denominator.

$- \frac{3 q^{3} p}{10} - \frac{3 q^{3} p}{5} + \frac{p q \cdot 2 - 7 p q}{10} + 3 p q$

Step 3

Combine the numerators over the common denominator.

$3 p q + \frac{- 3 q^{3} p + p q \cdot 2 - 7 p q}{10} + \frac{- 3 q^{3} p}{5}$

Step 4

Move $2$ to the left of $p q$ .

$3 p q + \frac{- 3 q^{3} p + 2 p q - 7 p q}{10} + \frac{- 3 q^{3} p}{5}$

Step 5

Subtract $7 p q$ from $2 p q$ .

$3 p q + \frac{- 3 q^{3} p - 5 p q}{10} + \frac{- 3 q^{3} p}{5}$

Step 6

Simplify each term.

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Step 6.1

Factor $q p$ out of $- 3 q^{3} p - 5 p q$ .

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Step 6.1.1

Factor $q p$ out of $- 3 q^{3} p$ .

$3 p q + \frac{q p (- 3 q^{2}) - 5 p q}{10} + \frac{- 3 q^{3} p}{5}$

Step 6.1.2

Factor $q p$ out of $- 5 p q$ .

$3 p q + \frac{q p (- 3 q^{2}) + q p (- 5)}{10} + \frac{- 3 q^{3} p}{5}$

Step 6.1.3

Factor $q p$ out of $q p (- 3 q^{2}) + q p (- 5)$ .

$3 p q + \frac{q p (- 3 q^{2} - 5)}{10} + \frac{- 3 q^{3} p}{5}$

Step 6.2

Move the negative in front of the fraction.

$3 p q + \frac{q p (- 3 q^{2} - 5)}{10} - \frac{3 q^{3} p}{5}$

Step 7

To write $3 p q$ as a fraction with a common denominator, multiply by $\frac{10}{10}$ .

$3 p q \cdot \frac{10}{10} + \frac{q p (- 3 q^{2} - 5)}{10} - \frac{3 q^{3} p}{5}$

Step 8

Simplify terms.

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Step 8.1

Combine $3 p q$ and $\frac{10}{10}$ .

$\frac{3 p q \cdot 10}{10} + \frac{q p (- 3 q^{2} - 5)}{10} - \frac{3 q^{3} p}{5}$

Step 8.2

Combine the numerators over the common denominator.

$\frac{3 p q \cdot 10 + q p (- 3 q^{2} - 5)}{10} - \frac{3 q^{3} p}{5}$

Step 9

Simplify the numerator.

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Step 9.1

Factor $p q$ out of $3 p q \cdot 10 + q p (- 3 q^{2} - 5)$ .

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Step 9.1.1

Factor $p q$ out of $3 p q \cdot 10$ .

$\frac{p q (3 \cdot 10) + q p (- 3 q^{2} - 5)}{10} - \frac{3 q^{3} p}{5}$

Step 9.1.2

Factor $p q$ out of $q p (- 3 q^{2} - 5)$ .

$\frac{p q (3 \cdot 10) + p q (- 3 q^{2} - 5)}{10} - \frac{3 q^{3} p}{5}$

Step 9.1.3

Factor $p q$ out of $p q (3 \cdot 10) + p q (- 3 q^{2} - 5)$ .

$\frac{p q (3 \cdot 10 - 3 q^{2} - 5)}{10} - \frac{3 q^{3} p}{5}$

Step 9.2

Multiply $3$ by $10$ .

$\frac{p q (30 - 3 q^{2} - 5)}{10} - \frac{3 q^{3} p}{5}$

Step 9.3

Subtract $5$ from $30$ .

$\frac{p q (- 3 q^{2} + 25)}{10} - \frac{3 q^{3} p}{5}$

Step 10

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

$\frac{p q (- 3 q^{2} + 25)}{10} - \frac{3 q^{3} p}{5} \cdot \frac{2}{2}$

Step 11

Write each expression with a common denominator of $10$ , by multiplying each by an appropriate factor of $1$ .

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Step 11.1

Multiply $\frac{3 q^{3} p}{5}$ by $\frac{2}{2}$ .

$\frac{p q (- 3 q^{2} + 25)}{10} - \frac{3 q^{3} p \cdot 2}{5 \cdot 2}$

Step 11.2

Multiply $5$ by $2$ .

$\frac{p q (- 3 q^{2} + 25)}{10} - \frac{3 q^{3} p \cdot 2}{10}$

Step 12

Combine the numerators over the common denominator.

$\frac{p q (- 3 q^{2} + 25) - 3 q^{3} p \cdot 2}{10}$

Step 13

Simplify the numerator.

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Step 13.1

Factor $p q$ out of $p q (- 3 q^{2} + 25) - 3 q^{3} p \cdot 2$ .

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Step 13.1.1

Factor $p q$ out of $- 3 q^{3} p \cdot 2$ .

$\frac{p q (- 3 q^{2} + 25) + p q (- 3 q^{2} \cdot 2)}{10}$

Step 13.1.2

Factor $p q$ out of $p q (- 3 q^{2} + 25) + p q (- 3 q^{2} \cdot 2)$ .

$\frac{p q (- 3 q^{2} + 25 - 3 q^{2} \cdot 2)}{10}$

Step 13.2

Multiply $2$ by $- 3$ .

$\frac{p q (- 3 q^{2} + 25 - 6 q^{2})}{10}$

Step 13.3

Subtract $6 q^{2}$ from $- 3 q^{2}$ .

$\frac{p q (- 9 q^{2} + 25)}{10}$

Step 13.4

Rewrite $25$ as $5^{2}$ .

$\frac{p q (- 9 q^{2} + 5^{2})}{10}$

Step 13.5

Rewrite $9 q^{2}$ as ${(3 q)}^{2}$ .

$\frac{p q (- {(3 q)}^{2} + 5^{2})}{10}$

Step 13.6

Reorder $- {(3 q)}^{2}$ and $5^{2}$ .

$\frac{p q (5^{2} - {(3 q)}^{2})}{10}$

Step 13.7

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 5$ and $b = 3 q$ .

$\frac{p q ((5 + 3 q) (5 - (3 q)))}{10}$

Step 13.8

Multiply $3$ by $- 1$ .

$\frac{p q (5 + 3 q) (5 - 3 q)}{10}$