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

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

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

To write

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

$\frac{2}{2}$ .

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

Step 3

Write each expression with a common denominator of

$10$ , by multiplying each by an appropriate factor of

$1$ .

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Step 3.1

Multiply

$\frac{p q}{5}$ by

$\frac{2}{2}$ .

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

Step 3.2

Multiply

$5$ by

$2$ .

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

Step 4

Simplify terms.

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Step 4.1

Combine the numerators over the common denominator.

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

Step 4.2

Combine the numerators over the common denominator.

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

Step 5

Move

$2$ to the left of

$p q$ .

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

Step 6

Subtract

$7 q p$ from

$2 p q$ .

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Step 6.1

Move

$q$ .

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

Step 6.2

Subtract

$7 p q$ from

$2 p q$ .

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

Step 7

Simplify each term.

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Step 7.1

Factor

$p q$ out of

$- 5 p q - 3 q^{3} p$ .

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Step 7.1.1

Factor

$p q$ out of

$- 5 p q$ .

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

Step 7.1.2

Factor

$p q$ out of

$- 3 q^{3} p$ .

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

Step 7.1.3

Factor

$p q$ out of

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

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

Step 7.2

Move the negative in front of the fraction.

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

Step 8

To write

$3 p q$ as a fraction with a common denominator, multiply by

$\frac{10}{10}$ .

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

Step 9

Simplify terms.

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Step 9.1

Combine

$3 p q$ and

$\frac{10}{10}$ .

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

Step 9.2

Combine the numerators over the common denominator.

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

Step 10

Simplify the numerator.

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Step 10.1

Factor

$p q$ out of

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

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Step 10.1.1

Factor

$p q$ out of

$3 p q \cdot 10$ .

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

Step 10.1.2

Factor

$p q$ out of

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

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

Step 10.2

Multiply

$3$ by

$10$ .

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

Step 10.3

Subtract

$5$ from

$30$ .

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

Step 11

To write

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

$\frac{2}{2}$ .

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

Step 12

Write each expression with a common denominator of

$10$ , by multiplying each by an appropriate factor of

$1$ .

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Step 12.1

Multiply

$\frac{3 q^{3} p}{5}$ by

$\frac{2}{2}$ .

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

Step 12.2

Multiply

$5$ by

$2$ .

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

Step 13

Combine the numerators over the common denominator.

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

Step 14

Simplify the numerator.

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Step 14.1

Factor

$p q$ out of

$p q (25 - 3 q^{2}) - 3 q^{3} p \cdot 2$ .

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Step 14.1.1

Factor

$p q$ out of

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

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

Step 14.1.2

Factor

$p q$ out of

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

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

Step 14.2

Multiply

$2$ by

$- 3$ .

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

Step 14.3

Subtract

$6 q^{2}$ from

$- 3 q^{2}$ .

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

Step 14.4

Rewrite

$25$ as

$5^{2}$ .

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

Step 14.5

Rewrite

$9 q^{2}$ as

${(3 q)}^{2}$ .

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

Step 14.6

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 14.7

Multiply

$3$ by

$- 1$ .

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