Pre-Algebra Examples

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

Step 1

Simplify terms.

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Step 1.1

Simplify each term.

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Step 1.1.1

Apply the distributive property.

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

Step 1.1.2

Combine $\frac{5}{7}$ and $p^{3}$ .

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

Step 1.1.3

Multiply $\frac{5}{7} \cdot - 2$ .

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Step 1.1.3.1

Combine $\frac{5}{7}$ and $- 2$ .

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

Step 1.1.3.2

Multiply $5$ by $- 2$ .

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

Step 1.1.4

Move the negative in front of the fraction.

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

Step 1.2

Rewrite ${(\frac{5 p^{3}}{7} - \frac{10}{7} + 7 q^{2})}^{2}$ as $(\frac{5 p^{3}}{7} - \frac{10}{7} + 7 q^{2}) (\frac{5 p^{3}}{7} - \frac{10}{7} + 7 q^{2})$ .

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

Step 2

Expand $(\frac{5 p^{3}}{7} - \frac{10}{7} + 7 q^{2}) (\frac{5 p^{3}}{7} - \frac{10}{7} + 7 q^{2})$ by multiplying each term in the first expression by each term in the second expression.

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

Step 3

Simplify terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Combine.

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

Step 3.1.2

Multiply $p^{3}$ by $p^{3}$ by adding the exponents.

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Step 3.1.2.1

Move $p^{3}$ .

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

Step 3.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 3.1.2.3

Add $3$ and $3$ .

$\frac{5 p^{6} \cdot 5}{7 \cdot 7} + \frac{5 p^{3}}{7} (- \frac{10}{7}) + \frac{5 p^{3}}{7} (7 q^{2}) - \frac{10}{7} \cdot \frac{5 p^{3}}{7} - \frac{10}{7} (- \frac{10}{7}) - \frac{10}{7} (7 q^{2}) + 7 q^{2} \frac{5 p^{3}}{7} + 7 q^{2} (- \frac{10}{7}) + 7 q^{2} (7 q^{2})$

Step 3.1.3

Multiply $5$ by $5$ .

$\frac{25 p^{6}}{7 \cdot 7} + \frac{5 p^{3}}{7} (- \frac{10}{7}) + \frac{5 p^{3}}{7} (7 q^{2}) - \frac{10}{7} \cdot \frac{5 p^{3}}{7} - \frac{10}{7} (- \frac{10}{7}) - \frac{10}{7} (7 q^{2}) + 7 q^{2} \frac{5 p^{3}}{7} + 7 q^{2} (- \frac{10}{7}) + 7 q^{2} (7 q^{2})$

Step 3.1.4

Multiply $7$ by $7$ .

$\frac{25 p^{6}}{49} + \frac{5 p^{3}}{7} (- \frac{10}{7}) + \frac{5 p^{3}}{7} (7 q^{2}) - \frac{10}{7} \cdot \frac{5 p^{3}}{7} - \frac{10}{7} (- \frac{10}{7}) - \frac{10}{7} (7 q^{2}) + 7 q^{2} \frac{5 p^{3}}{7} + 7 q^{2} (- \frac{10}{7}) + 7 q^{2} (7 q^{2})$

Step 3.1.5

Multiply $\frac{5 p^{3}}{7} (- \frac{10}{7})$ .

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Step 3.1.5.1

Multiply $\frac{5 p^{3}}{7}$ by $\frac{10}{7}$ .

$\frac{25 p^{6}}{49} - \frac{5 p^{3} \cdot 10}{7 \cdot 7} + \frac{5 p^{3}}{7} (7 q^{2}) - \frac{10}{7} \cdot \frac{5 p^{3}}{7} - \frac{10}{7} (- \frac{10}{7}) - \frac{10}{7} (7 q^{2}) + 7 q^{2} \frac{5 p^{3}}{7} + 7 q^{2} (- \frac{10}{7}) + 7 q^{2} (7 q^{2})$

Step 3.1.5.2

Multiply $10$ by $5$ .

$\frac{25 p^{6}}{49} - \frac{50 p^{3}}{7 \cdot 7} + \frac{5 p^{3}}{7} (7 q^{2}) - \frac{10}{7} \cdot \frac{5 p^{3}}{7} - \frac{10}{7} (- \frac{10}{7}) - \frac{10}{7} (7 q^{2}) + 7 q^{2} \frac{5 p^{3}}{7} + 7 q^{2} (- \frac{10}{7}) + 7 q^{2} (7 q^{2})$

Step 3.1.5.3

Multiply $7$ by $7$ .

$\frac{25 p^{6}}{49} - \frac{50 p^{3}}{49} + \frac{5 p^{3}}{7} (7 q^{2}) - \frac{10}{7} \cdot \frac{5 p^{3}}{7} - \frac{10}{7} (- \frac{10}{7}) - \frac{10}{7} (7 q^{2}) + 7 q^{2} \frac{5 p^{3}}{7} + 7 q^{2} (- \frac{10}{7}) + 7 q^{2} (7 q^{2})$

Step 3.1.6

Rewrite using the commutative property of multiplication.

$\frac{25 p^{6}}{49} - \frac{50 p^{3}}{49} + 7 \frac{5 p^{3}}{7} q^{2} - \frac{10}{7} \cdot \frac{5 p^{3}}{7} - \frac{10}{7} (- \frac{10}{7}) - \frac{10}{7} (7 q^{2}) + 7 q^{2} \frac{5 p^{3}}{7} + 7 q^{2} (- \frac{10}{7}) + 7 q^{2} (7 q^{2})$

Step 3.1.7

Cancel the common factor of $7$ .

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Step 3.1.7.1

Cancel the common factor.

Step 3.1.7.2

Rewrite the expression.

$\frac{25 p^{6}}{49} - \frac{50 p^{3}}{49} + 5 p^{3} q^{2} - \frac{10}{7} \cdot \frac{5 p^{3}}{7} - \frac{10}{7} (- \frac{10}{7}) - \frac{10}{7} (7 q^{2}) + 7 q^{2} \frac{5 p^{3}}{7} + 7 q^{2} (- \frac{10}{7}) + 7 q^{2} (7 q^{2})$

Step 3.1.8

Multiply $- \frac{10}{7} \cdot \frac{5 p^{3}}{7}$ .

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Step 3.1.8.1

Multiply $\frac{5 p^{3}}{7}$ by $\frac{10}{7}$ .

$\frac{25 p^{6}}{49} - \frac{50 p^{3}}{49} + 5 p^{3} q^{2} - \frac{5 p^{3} \cdot 10}{7 \cdot 7} - \frac{10}{7} (- \frac{10}{7}) - \frac{10}{7} (7 q^{2}) + 7 q^{2} \frac{5 p^{3}}{7} + 7 q^{2} (- \frac{10}{7}) + 7 q^{2} (7 q^{2})$

Step 3.1.8.2

Multiply $10$ by $5$ .

$\frac{25 p^{6}}{49} - \frac{50 p^{3}}{49} + 5 p^{3} q^{2} - \frac{50 p^{3}}{7 \cdot 7} - \frac{10}{7} (- \frac{10}{7}) - \frac{10}{7} (7 q^{2}) + 7 q^{2} \frac{5 p^{3}}{7} + 7 q^{2} (- \frac{10}{7}) + 7 q^{2} (7 q^{2})$

Step 3.1.8.3

Multiply $7$ by $7$ .

$\frac{25 p^{6}}{49} - \frac{50 p^{3}}{49} + 5 p^{3} q^{2} - \frac{50 p^{3}}{49} - \frac{10}{7} (- \frac{10}{7}) - \frac{10}{7} (7 q^{2}) + 7 q^{2} \frac{5 p^{3}}{7} + 7 q^{2} (- \frac{10}{7}) + 7 q^{2} (7 q^{2})$

Step 3.1.9

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

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Step 3.1.9.1

Multiply $- 1$ by $- 1$ .

$\frac{25 p^{6}}{49} - \frac{50 p^{3}}{49} + 5 p^{3} q^{2} - \frac{50 p^{3}}{49} + 1 (\frac{10}{7}) \frac{10}{7} - \frac{10}{7} (7 q^{2}) + 7 q^{2} \frac{5 p^{3}}{7} + 7 q^{2} (- \frac{10}{7}) + 7 q^{2} (7 q^{2})$

Step 3.1.9.2

Multiply $\frac{10}{7}$ by $1$ .

$\frac{25 p^{6}}{49} - \frac{50 p^{3}}{49} + 5 p^{3} q^{2} - \frac{50 p^{3}}{49} + \frac{10}{7} \cdot \frac{10}{7} - \frac{10}{7} (7 q^{2}) + 7 q^{2} \frac{5 p^{3}}{7} + 7 q^{2} (- \frac{10}{7}) + 7 q^{2} (7 q^{2})$

Step 3.1.9.3

Multiply $\frac{10}{7}$ by $\frac{10}{7}$ .

$\frac{25 p^{6}}{49} - \frac{50 p^{3}}{49} + 5 p^{3} q^{2} - \frac{50 p^{3}}{49} + \frac{10 \cdot 10}{7 \cdot 7} - \frac{10}{7} (7 q^{2}) + 7 q^{2} \frac{5 p^{3}}{7} + 7 q^{2} (- \frac{10}{7}) + 7 q^{2} (7 q^{2})$

Step 3.1.9.4

Multiply $10$ by $10$ .

$\frac{25 p^{6}}{49} - \frac{50 p^{3}}{49} + 5 p^{3} q^{2} - \frac{50 p^{3}}{49} + \frac{100}{7 \cdot 7} - \frac{10}{7} (7 q^{2}) + 7 q^{2} \frac{5 p^{3}}{7} + 7 q^{2} (- \frac{10}{7}) + 7 q^{2} (7 q^{2})$

Step 3.1.9.5

Multiply $7$ by $7$ .

$\frac{25 p^{6}}{49} - \frac{50 p^{3}}{49} + 5 p^{3} q^{2} - \frac{50 p^{3}}{49} + \frac{100}{49} - \frac{10}{7} (7 q^{2}) + 7 q^{2} \frac{5 p^{3}}{7} + 7 q^{2} (- \frac{10}{7}) + 7 q^{2} (7 q^{2})$

Step 3.1.10

Cancel the common factor of $7$ .

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Step 3.1.10.1

Move the leading negative in $- \frac{10}{7}$ into the numerator.

$\frac{25 p^{6}}{49} - \frac{50 p^{3}}{49} + 5 p^{3} q^{2} - \frac{50 p^{3}}{49} + \frac{100}{49} + \frac{- 10}{7} (7 q^{2}) + 7 q^{2} \frac{5 p^{3}}{7} + 7 q^{2} (- \frac{10}{7}) + 7 q^{2} (7 q^{2})$

Step 3.1.10.2

Factor $7$ out of $7 q^{2}$ .

$\frac{25 p^{6}}{49} - \frac{50 p^{3}}{49} + 5 p^{3} q^{2} - \frac{50 p^{3}}{49} + \frac{100}{49} + \frac{- 10}{7} (7 (q^{2})) + 7 q^{2} \frac{5 p^{3}}{7} + 7 q^{2} (- \frac{10}{7}) + 7 q^{2} (7 q^{2})$

Step 3.1.10.3

Cancel the common factor.

Step 3.1.10.4

Rewrite the expression.

$\frac{25 p^{6}}{49} - \frac{50 p^{3}}{49} + 5 p^{3} q^{2} - \frac{50 p^{3}}{49} + \frac{100}{49} - 10 q^{2} + 7 q^{2} \frac{5 p^{3}}{7} + 7 q^{2} (- \frac{10}{7}) + 7 q^{2} (7 q^{2})$

Step 3.1.11

Cancel the common factor of $7$ .

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Step 3.1.11.1

Factor $7$ out of $7 q^{2}$ .

$\frac{25 p^{6}}{49} - \frac{50 p^{3}}{49} + 5 p^{3} q^{2} - \frac{50 p^{3}}{49} + \frac{100}{49} - 10 q^{2} + 7 (q^{2}) \frac{5 p^{3}}{7} + 7 q^{2} (- \frac{10}{7}) + 7 q^{2} (7 q^{2})$

Step 3.1.11.2

Cancel the common factor.

$\frac{25 p^{6}}{49} - \frac{50 p^{3}}{49} + 5 p^{3} q^{2} - \frac{50 p^{3}}{49} + \frac{100}{49} - 10 q^{2} + 7 q^{2} \frac{5 p^{3}}{7} + 7 q^{2} (- \frac{10}{7}) + 7 q^{2} (7 q^{2})$

Step 3.1.11.3

Rewrite the expression.

$\frac{25 p^{6}}{49} - \frac{50 p^{3}}{49} + 5 p^{3} q^{2} - \frac{50 p^{3}}{49} + \frac{100}{49} - 10 q^{2} + q^{2} (5 p^{3}) + 7 q^{2} (- \frac{10}{7}) + 7 q^{2} (7 q^{2})$

Step 3.1.12

Rewrite using the commutative property of multiplication.

$\frac{25 p^{6}}{49} - \frac{50 p^{3}}{49} + 5 p^{3} q^{2} - \frac{50 p^{3}}{49} + \frac{100}{49} - 10 q^{2} + 5 q^{2} p^{3} + 7 q^{2} (- \frac{10}{7}) + 7 q^{2} (7 q^{2})$

Step 3.1.13

Cancel the common factor of $7$ .

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Step 3.1.13.1

Move the leading negative in $- \frac{10}{7}$ into the numerator.

$\frac{25 p^{6}}{49} - \frac{50 p^{3}}{49} + 5 p^{3} q^{2} - \frac{50 p^{3}}{49} + \frac{100}{49} - 10 q^{2} + 5 q^{2} p^{3} + 7 q^{2} \frac{- 10}{7} + 7 q^{2} (7 q^{2})$

Step 3.1.13.2

Factor $7$ out of $7 q^{2}$ .

$\frac{25 p^{6}}{49} - \frac{50 p^{3}}{49} + 5 p^{3} q^{2} - \frac{50 p^{3}}{49} + \frac{100}{49} - 10 q^{2} + 5 q^{2} p^{3} + 7 (q^{2}) \frac{- 10}{7} + 7 q^{2} (7 q^{2})$

Step 3.1.13.3

Cancel the common factor.

$\frac{25 p^{6}}{49} - \frac{50 p^{3}}{49} + 5 p^{3} q^{2} - \frac{50 p^{3}}{49} + \frac{100}{49} - 10 q^{2} + 5 q^{2} p^{3} + 7 q^{2} \frac{- 10}{7} + 7 q^{2} (7 q^{2})$

Step 3.1.13.4

Rewrite the expression.

$\frac{25 p^{6}}{49} - \frac{50 p^{3}}{49} + 5 p^{3} q^{2} - \frac{50 p^{3}}{49} + \frac{100}{49} - 10 q^{2} + 5 q^{2} p^{3} + q^{2} \cdot - 10 + 7 q^{2} (7 q^{2})$

Step 3.1.14

Move $- 10$ to the left of $q^{2}$ .

$\frac{25 p^{6}}{49} - \frac{50 p^{3}}{49} + 5 p^{3} q^{2} - \frac{50 p^{3}}{49} + \frac{100}{49} - 10 q^{2} + 5 q^{2} p^{3} - 10 \cdot q^{2} + 7 q^{2} (7 q^{2})$

Step 3.1.15

Rewrite using the commutative property of multiplication.

$\frac{25 p^{6}}{49} - \frac{50 p^{3}}{49} + 5 p^{3} q^{2} - \frac{50 p^{3}}{49} + \frac{100}{49} - 10 q^{2} + 5 q^{2} p^{3} - 10 q^{2} + 7 \cdot 7 q^{2} q^{2}$

Step 3.1.16

Multiply $q^{2}$ by $q^{2}$ by adding the exponents.

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Step 3.1.16.1

Move $q^{2}$ .

$\frac{25 p^{6}}{49} - \frac{50 p^{3}}{49} + 5 p^{3} q^{2} - \frac{50 p^{3}}{49} + \frac{100}{49} - 10 q^{2} + 5 q^{2} p^{3} - 10 q^{2} + 7 \cdot 7 (q^{2} q^{2})$

Step 3.1.16.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{25 p^{6}}{49} - \frac{50 p^{3}}{49} + 5 p^{3} q^{2} - \frac{50 p^{3}}{49} + \frac{100}{49} - 10 q^{2} + 5 q^{2} p^{3} - 10 q^{2} + 7 \cdot 7 q^{2 + 2}$

Step 3.1.16.3

Add $2$ and $2$ .

$\frac{25 p^{6}}{49} - \frac{50 p^{3}}{49} + 5 p^{3} q^{2} - \frac{50 p^{3}}{49} + \frac{100}{49} - 10 q^{2} + 5 q^{2} p^{3} - 10 q^{2} + 7 \cdot 7 q^{4}$

Step 3.1.17

Multiply $7$ by $7$ .

$\frac{25 p^{6}}{49} - \frac{50 p^{3}}{49} + 5 p^{3} q^{2} - \frac{50 p^{3}}{49} + \frac{100}{49} - 10 q^{2} + 5 q^{2} p^{3} - 10 q^{2} + 49 q^{4}$

Step 3.2

Simplify terms.

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Step 3.2.1

Combine the numerators over the common denominator.

$5 p^{3} q^{2} - 10 q^{2} + 5 q^{2} p^{3} - 10 q^{2} + 49 q^{4} + \frac{25 p^{6} - 50 p^{3} - 50 p^{3} + 100}{49}$

Step 3.2.2

Subtract $50 p^{3}$ from $- 50 p^{3}$ .

$5 p^{3} q^{2} - 10 q^{2} + 5 q^{2} p^{3} - 10 q^{2} + 49 q^{4} + \frac{25 p^{6} - 100 p^{3} + 100}{49}$

Step 4

Simplify the numerator.

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Step 4.1

Factor $25$ out of $25 p^{6} - 100 p^{3} + 100$ .

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Step 4.1.1

Factor $25$ out of $25 p^{6}$ .

$5 p^{3} q^{2} - 10 q^{2} + 5 q^{2} p^{3} - 10 q^{2} + 49 q^{4} + \frac{25 (p^{6}) - 100 p^{3} + 100}{49}$

Step 4.1.2

Factor $25$ out of $- 100 p^{3}$ .

$5 p^{3} q^{2} - 10 q^{2} + 5 q^{2} p^{3} - 10 q^{2} + 49 q^{4} + \frac{25 (p^{6}) + 25 (- 4 p^{3}) + 100}{49}$

Step 4.1.3

Factor $25$ out of $100$ .

$5 p^{3} q^{2} - 10 q^{2} + 5 q^{2} p^{3} - 10 q^{2} + 49 q^{4} + \frac{25 p^{6} + 25 (- 4 p^{3}) + 25 \cdot 4}{49}$

Step 4.1.4

Factor $25$ out of $25 p^{6} + 25 (- 4 p^{3})$ .

$5 p^{3} q^{2} - 10 q^{2} + 5 q^{2} p^{3} - 10 q^{2} + 49 q^{4} + \frac{25 (p^{6} - 4 p^{3}) + 25 \cdot 4}{49}$

Step 4.1.5

Factor $25$ out of $25 (p^{6} - 4 p^{3}) + 25 \cdot 4$ .

$5 p^{3} q^{2} - 10 q^{2} + 5 q^{2} p^{3} - 10 q^{2} + 49 q^{4} + \frac{25 (p^{6} - 4 p^{3} + 4)}{49}$

Step 4.2

Rewrite $p^{6}$ as ${(p^{3})}^{2}$ .

$5 p^{3} q^{2} - 10 q^{2} + 5 q^{2} p^{3} - 10 q^{2} + 49 q^{4} + \frac{25 ({(p^{3})}^{2} - 4 p^{3} + 4)}{49}$

Step 4.3

Let $u = p^{3}$ . Substitute $u$ for all occurrences of $p^{3}$ .

$5 p^{3} q^{2} - 10 q^{2} + 5 q^{2} p^{3} - 10 q^{2} + 49 q^{4} + \frac{25 (u^{2} - 4 u + 4)}{49}$

Step 4.4

Factor using the perfect square rule.

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Step 4.4.1

Rewrite $4$ as $2^{2}$ .

$5 p^{3} q^{2} - 10 q^{2} + 5 q^{2} p^{3} - 10 q^{2} + 49 q^{4} + \frac{25 (u^{2} - 4 u + 2^{2})}{49}$

Step 4.4.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$4 u = 2 \cdot u \cdot 2$

Step 4.4.3

Rewrite the polynomial.

$5 p^{3} q^{2} - 10 q^{2} + 5 q^{2} p^{3} - 10 q^{2} + 49 q^{4} + \frac{25 (u^{2} - 2 \cdot u \cdot 2 + 2^{2})}{49}$

Step 4.4.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = u$ and $b = 2$ .

$5 p^{3} q^{2} - 10 q^{2} + 5 q^{2} p^{3} - 10 q^{2} + 49 q^{4} + \frac{25 {(u - 2)}^{2}}{49}$

Step 4.5

Replace all occurrences of $u$ with $p^{3}$ .

$5 p^{3} q^{2} - 10 q^{2} + 5 q^{2} p^{3} - 10 q^{2} + 49 q^{4} + \frac{25 {(p^{3} - 2)}^{2}}{49}$

Step 5

Add $5 p^{3} q^{2}$ and $5 q^{2} p^{3}$ .

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Step 5.1

Move $q^{2}$ .

$- 10 q^{2} + 5 p^{3} q^{2} + 5 p^{3} q^{2} - 10 q^{2} + 49 q^{4} + \frac{25 {(p^{3} - 2)}^{2}}{49}$

Step 5.2

Add $5 p^{3} q^{2}$ and $5 p^{3} q^{2}$ .

$- 10 q^{2} + 10 p^{3} q^{2} - 10 q^{2} + 49 q^{4} + \frac{25 {(p^{3} - 2)}^{2}}{49}$

Step 6

Subtract $10 q^{2}$ from $- 10 q^{2}$ .

$- 20 q^{2} + 10 p^{3} q^{2} + 49 q^{4} + \frac{25 {(p^{3} - 2)}^{2}}{49}$

Step 7

To write $- 20 q^{2}$ as a fraction with a common denominator, multiply by $\frac{49}{49}$ .

$10 p^{3} q^{2} + 49 q^{4} - 20 q^{2} \cdot \frac{49}{49} + \frac{25 {(p^{3} - 2)}^{2}}{49}$

Step 8

Simplify terms.

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Step 8.1

Combine $- 20 q^{2}$ and $\frac{49}{49}$ .

$10 p^{3} q^{2} + 49 q^{4} + \frac{- 20 q^{2} \cdot 49}{49} + \frac{25 {(p^{3} - 2)}^{2}}{49}$

Step 8.2

Combine the numerators over the common denominator.

$10 p^{3} q^{2} + 49 q^{4} + \frac{- 20 q^{2} \cdot 49 + 25 {(p^{3} - 2)}^{2}}{49}$

Step 9

Simplify the numerator.

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Step 9.1

Factor $5$ out of $- 20 q^{2} \cdot 49 + 25 {(p^{3} - 2)}^{2}$ .

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Step 9.1.1

Factor $5$ out of $- 20 q^{2} \cdot 49$ .

$10 p^{3} q^{2} + 49 q^{4} + \frac{5 (- 4 q^{2} \cdot 49) + 25 {(p^{3} - 2)}^{2}}{49}$

Step 9.1.2

Factor $5$ out of $25 {(p^{3} - 2)}^{2}$ .

$10 p^{3} q^{2} + 49 q^{4} + \frac{5 (- 4 q^{2} \cdot 49) + 5 (5 {(p^{3} - 2)}^{2})}{49}$

Step 9.1.3

Factor $5$ out of $5 (- 4 q^{2} \cdot 49) + 5 (5 {(p^{3} - 2)}^{2})$ .

$10 p^{3} q^{2} + 49 q^{4} + \frac{5 (- 4 q^{2} \cdot 49 + 5 {(p^{3} - 2)}^{2})}{49}$

Step 9.2

Multiply $49$ by $- 4$ .

$10 p^{3} q^{2} + 49 q^{4} + \frac{5 (- 196 q^{2} + 5 {(p^{3} - 2)}^{2})}{49}$

Step 9.3

Rewrite ${(p^{3} - 2)}^{2}$ as $(p^{3} - 2) (p^{3} - 2)$ .

$10 p^{3} q^{2} + 49 q^{4} + \frac{5 (- 196 q^{2} + 5 ((p^{3} - 2) (p^{3} - 2)))}{49}$

Step 9.4

Expand $(p^{3} - 2) (p^{3} - 2)$ using the FOIL Method.

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Step 9.4.1

Apply the distributive property.

$10 p^{3} q^{2} + 49 q^{4} + \frac{5 (- 196 q^{2} + 5 (p^{3} (p^{3} - 2) - 2 (p^{3} - 2)))}{49}$

Step 9.4.2

Apply the distributive property.

$10 p^{3} q^{2} + 49 q^{4} + \frac{5 (- 196 q^{2} + 5 (p^{3} p^{3} + p^{3} \cdot - 2 - 2 (p^{3} - 2)))}{49}$

Step 9.4.3

Apply the distributive property.

$10 p^{3} q^{2} + 49 q^{4} + \frac{5 (- 196 q^{2} + 5 (p^{3} p^{3} + p^{3} \cdot - 2 - 2 p^{3} - 2 \cdot - 2))}{49}$

Step 9.5

Simplify and combine like terms.

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Step 9.5.1

Simplify each term.

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Step 9.5.1.1

Multiply $p^{3}$ by $p^{3}$ by adding the exponents.

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Step 9.5.1.1.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$10 p^{3} q^{2} + 49 q^{4} + \frac{5 (- 196 q^{2} + 5 (p^{3 + 3} + p^{3} \cdot - 2 - 2 p^{3} - 2 \cdot - 2))}{49}$

Step 9.5.1.1.2

Add $3$ and $3$ .

$10 p^{3} q^{2} + 49 q^{4} + \frac{5 (- 196 q^{2} + 5 (p^{6} + p^{3} \cdot - 2 - 2 p^{3} - 2 \cdot - 2))}{49}$

Step 9.5.1.2

Move $- 2$ to the left of $p^{3}$ .

$10 p^{3} q^{2} + 49 q^{4} + \frac{5 (- 196 q^{2} + 5 (p^{6} - 2 \cdot p^{3} - 2 p^{3} - 2 \cdot - 2))}{49}$

Step 9.5.1.3

Multiply $- 2$ by $- 2$ .

$10 p^{3} q^{2} + 49 q^{4} + \frac{5 (- 196 q^{2} + 5 (p^{6} - 2 p^{3} - 2 p^{3} + 4))}{49}$

Step 9.5.2

Subtract $2 p^{3}$ from $- 2 p^{3}$ .

$10 p^{3} q^{2} + 49 q^{4} + \frac{5 (- 196 q^{2} + 5 (p^{6} - 4 p^{3} + 4))}{49}$

Step 9.6

Apply the distributive property.

$10 p^{3} q^{2} + 49 q^{4} + \frac{5 (- 196 q^{2} + 5 p^{6} + 5 (- 4 p^{3}) + 5 \cdot 4)}{49}$

Step 9.7

Simplify.

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Step 9.7.1

Multiply $- 4$ by $5$ .

$10 p^{3} q^{2} + 49 q^{4} + \frac{5 (- 196 q^{2} + 5 p^{6} - 20 p^{3} + 5 \cdot 4)}{49}$

Step 9.7.2

Multiply $5$ by $4$ .

$10 p^{3} q^{2} + 49 q^{4} + \frac{5 (- 196 q^{2} + 5 p^{6} - 20 p^{3} + 20)}{49}$

Step 10

To write $10 p^{3} q^{2}$ as a fraction with a common denominator, multiply by $\frac{49}{49}$ .

$49 q^{4} + 10 p^{3} q^{2} \cdot \frac{49}{49} + \frac{5 (- 196 q^{2} + 5 p^{6} - 20 p^{3} + 20)}{49}$

Step 11

Simplify terms.

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Step 11.1

Combine $10 p^{3} q^{2}$ and $\frac{49}{49}$ .

$49 q^{4} + \frac{10 p^{3} q^{2} \cdot 49}{49} + \frac{5 (- 196 q^{2} + 5 p^{6} - 20 p^{3} + 20)}{49}$

Step 11.2

Combine the numerators over the common denominator.

$49 q^{4} + \frac{10 p^{3} q^{2} \cdot 49 + 5 (- 196 q^{2} + 5 p^{6} - 20 p^{3} + 20)}{49}$

Step 12

Simplify the numerator.

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Step 12.1

Factor $5$ out of $10 p^{3} q^{2} \cdot 49 + 5 (- 196 q^{2} + 5 p^{6} - 20 p^{3} + 20)$ .

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Step 12.1.1

Factor $5$ out of $10 p^{3} q^{2} \cdot 49$ .

$49 q^{4} + \frac{5 (2 p^{3} q^{2} \cdot 49) + 5 (- 196 q^{2} + 5 p^{6} - 20 p^{3} + 20)}{49}$

Step 12.1.2

Factor $5$ out of $5 (2 p^{3} q^{2} \cdot 49) + 5 (- 196 q^{2} + 5 p^{6} - 20 p^{3} + 20)$ .

$49 q^{4} + \frac{5 (2 p^{3} q^{2} \cdot 49 - 196 q^{2} + 5 p^{6} - 20 p^{3} + 20)}{49}$

Step 12.2

Multiply $49$ by $2$ .

$49 q^{4} + \frac{5 (98 p^{3} q^{2} - 196 q^{2} + 5 p^{6} - 20 p^{3} + 20)}{49}$

Step 13

To write $49 q^{4}$ as a fraction with a common denominator, multiply by $\frac{49}{49}$ .

$49 q^{4} \cdot \frac{49}{49} + \frac{5 (98 p^{3} q^{2} - 196 q^{2} + 5 p^{6} - 20 p^{3} + 20)}{49}$

Step 14

Simplify terms.

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Step 14.1

Combine $49 q^{4}$ and $\frac{49}{49}$ .

$\frac{49 q^{4} \cdot 49}{49} + \frac{5 (98 p^{3} q^{2} - 196 q^{2} + 5 p^{6} - 20 p^{3} + 20)}{49}$

Step 14.2

Combine the numerators over the common denominator.

$\frac{49 q^{4} \cdot 49 + 5 (98 p^{3} q^{2} - 196 q^{2} + 5 p^{6} - 20 p^{3} + 20)}{49}$

Step 15

Simplify the numerator.

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Step 15.1

Multiply $49$ by $49$ .

$\frac{2401 q^{4} + 5 (98 p^{3} q^{2} - 196 q^{2} + 5 p^{6} - 20 p^{3} + 20)}{49}$

Step 15.2

Apply the distributive property.

$\frac{2401 q^{4} + 5 (98 p^{3} q^{2}) + 5 (- 196 q^{2}) + 5 (5 p^{6}) + 5 (- 20 p^{3}) + 5 \cdot 20}{49}$

Step 15.3

Simplify.

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Step 15.3.1

Multiply $98$ by $5$ .

$\frac{2401 q^{4} + 490 (p^{3} q^{2}) + 5 (- 196 q^{2}) + 5 (5 p^{6}) + 5 (- 20 p^{3}) + 5 \cdot 20}{49}$

Step 15.3.2

Multiply $- 196$ by $5$ .

$\frac{2401 q^{4} + 490 (p^{3} q^{2}) - 980 q^{2} + 5 (5 p^{6}) + 5 (- 20 p^{3}) + 5 \cdot 20}{49}$

Step 15.3.3

Multiply $5$ by $5$ .

$\frac{2401 q^{4} + 490 (p^{3} q^{2}) - 980 q^{2} + 25 p^{6} + 5 (- 20 p^{3}) + 5 \cdot 20}{49}$

Step 15.3.4

Multiply $- 20$ by $5$ .

$\frac{2401 q^{4} + 490 (p^{3} q^{2}) - 980 q^{2} + 25 p^{6} - 100 p^{3} + 5 \cdot 20}{49}$

Step 15.3.5

Multiply $5$ by $20$ .

$\frac{2401 q^{4} + 490 (p^{3} q^{2}) - 980 q^{2} + 25 p^{6} - 100 p^{3} + 100}{49}$

$\frac{2401 q^{4} + 490 p^{3} q^{2} - 980 q^{2} + 25 p^{6} - 100 p^{3} + 100}{49}$