Basic Math Examples

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

Step 1

To divide by a fraction, multiply by its reciprocal.

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

Step 2

Simplify each term.

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Step 2.1

Factor $q$ out of $p q + q^{2}$ .

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Step 2.1.1

Factor $q$ out of $p q$ .

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

Step 2.1.2

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

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

Step 2.1.3

Factor $q$ out of $q p + q \cdot q$ .

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

Step 2.2

Factor $p$ out of $p^{2} + p q$ .

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Step 2.2.1

Factor $p$ out of $p^{2}$ .

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

Step 2.2.2

Factor $p$ out of $p q$ .

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

Step 2.2.3

Factor $p$ out of $p \cdot p + p (q)$ .

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

Step 3

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

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

Step 4

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

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

Step 5

Write each expression with a common denominator of $p q (p + q)$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.1

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

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

Step 5.2

Multiply $\frac{p^{2}}{q (p + q)}$ by $\frac{p}{p}$ .

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

Step 5.3

Reorder the factors of $q (p + q) p$ .

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

Step 6

Combine the numerators over the common denominator.

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

Step 7

Simplify each term.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Expand $(p^{2} + q^{2}) (p + q)$ using the FOIL Method.

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Step 7.1.1.1

Apply the distributive property.

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

Step 7.1.1.2

Apply the distributive property.

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

Step 7.1.1.3

Apply the distributive property.

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

Step 7.1.2

Simplify each term.

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Step 7.1.2.1

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

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Step 7.1.2.1.1

Multiply $p^{2}$ by $p$ .

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Step 7.1.2.1.1.1

Raise $p$ to the power of $1$ .

$(\frac{p^{2} p^{1} + p^{2} q + q^{2} p + q^{2} q - p^{2} p}{p q (p + q)} - \frac{q^{2}}{p (p + q)}) \frac{p q}{3}$

Step 7.1.2.1.1.2

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

$(\frac{p^{2 + 1} + p^{2} q + q^{2} p + q^{2} q - p^{2} p}{p q (p + q)} - \frac{q^{2}}{p (p + q)}) \frac{p q}{3}$

Step 7.1.2.1.2

Add $2$ and $1$ .

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

Step 7.1.2.2

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

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Step 7.1.2.2.1

Multiply $q^{2}$ by $q$ .

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Step 7.1.2.2.1.1

Raise $q$ to the power of $1$ .

$(\frac{p^{3} + p^{2} q + q^{2} p + q^{2} q^{1} - p^{2} p}{p q (p + q)} - \frac{q^{2}}{p (p + q)}) \frac{p q}{3}$

Step 7.1.2.2.1.2

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

$(\frac{p^{3} + p^{2} q + q^{2} p + q^{2 + 1} - p^{2} p}{p q (p + q)} - \frac{q^{2}}{p (p + q)}) \frac{p q}{3}$

Step 7.1.2.2.2

Add $2$ and $1$ .

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

Step 7.1.3

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

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Step 7.1.3.1

Move $p$ .

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

Step 7.1.3.2

Multiply $p$ by $p^{2}$ .

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Step 7.1.3.2.1

Raise $p$ to the power of $1$ .

$(\frac{p^{3} + p^{2} q + q^{2} p + q^{3} - (p^{1} p^{2})}{p q (p + q)} - \frac{q^{2}}{p (p + q)}) \frac{p q}{3}$

Step 7.1.3.2.2

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

$(\frac{p^{3} + p^{2} q + q^{2} p + q^{3} - p^{1 + 2}}{p q (p + q)} - \frac{q^{2}}{p (p + q)}) \frac{p q}{3}$

Step 7.1.3.3

Add $1$ and $2$ .

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

Step 7.1.4

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

$(\frac{p^{2} q + q^{2} p + q^{3} + 0}{p q (p + q)} - \frac{q^{2}}{p (p + q)}) \frac{p q}{3}$

Step 7.1.5

Add $p^{2} q + q^{2} p + q^{3}$ and $0$ .

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

Step 7.1.6

Factor $q$ out of $p^{2} q + q^{2} p + q^{3}$ .

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Step 7.1.6.1

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

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

Step 7.1.6.2

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

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

Step 7.1.6.3

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

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

Step 7.1.6.4

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

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

Step 7.1.6.5

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

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

Step 7.2

Cancel the common factor of $q$ .

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Step 7.2.1

Cancel the common factor.

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

Step 7.2.2

Rewrite the expression.

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

Step 8

Simplify terms.

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Step 8.1

Combine the numerators over the common denominator.

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

Step 8.2

Combine the opposite terms in $p^{2} + q p + q^{2} - q^{2}$ .

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Step 8.2.1

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

$\frac{p^{2} + q p + 0}{p (p + q)} \cdot \frac{p q}{3}$

Step 8.2.2

Add $p^{2} + q p$ and $0$ .

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

Step 8.3

Factor $p$ out of $p^{2} + q p$ .

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Step 8.3.1

Factor $p$ out of $p^{2}$ .

$\frac{p \cdot p + q p}{p (p + q)} \cdot \frac{p q}{3}$

Step 8.3.2

Factor $p$ out of $q p$ .

$\frac{p \cdot p + p q}{p (p + q)} \cdot \frac{p q}{3}$

Step 8.3.3

Factor $p$ out of $p \cdot p + p q$ .

$\frac{p (p + q)}{p (p + q)} \cdot \frac{p q}{3}$

Step 9

Cancel the common factor.

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Step 9.1

Cancel the common factor of $p$ .

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Step 9.1.1

Cancel the common factor.

$\frac{p (p + q)}{p (p + q)} \cdot \frac{p q}{3}$

Step 9.1.2

Rewrite the expression.

$\frac{p + q}{p + q} \cdot \frac{p q}{3}$

Step 9.2

Cancel the common factor of $p + q$ .

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Step 9.2.1

Cancel the common factor.

$\frac{p + q}{p + q} \cdot \frac{p q}{3}$

Step 9.2.2

Rewrite the expression.

$1 \frac{p q}{3}$

Step 10

Multiply $\frac{p q}{3}$ by $1$ .

$\frac{p q}{3}$