Algebra Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Write each expression with a common denominator of $p^{3} q^{3}$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

Add $1$ and $2$ .

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

Step 3.5

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

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

Step 3.6

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

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

Step 3.7

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

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

Step 3.8

Add $1$ and $2$ .

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

Tap for more steps...

Step 5.1

Apply the distributive property.

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

Step 5.2

Multiply $q$ by $q$ by adding the exponents.

Tap for more steps...

Step 5.2.1

Move $q$ .

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

Step 5.2.2

Multiply $q$ by $q$ .

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

Step 5.3

Apply the distributive property.

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

Step 5.4

Apply the distributive property.

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

Step 5.5

Multiply $p$ by $p$ by adding the exponents.

Tap for more steps...

Step 5.5.1

Move $p$ .

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

Step 5.5.2

Multiply $p$ by $p$ .

$\frac{p q - q^{2} - p^{2} - q p}{p^{3} q^{3}}$

Step 5.6

Subtract $q p$ from $p q$ .

Tap for more steps...

Step 5.6.1

Move $q$ .

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

Step 5.6.2

Subtract $p q$ from $p q$ .

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

Step 5.7

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

$\frac{- q^{2} - p^{2}}{p^{3} q^{3}}$

Step 6

Simplify with factoring out.

Tap for more steps...

Step 6.1

Factor $- 1$ out of $- q^{2}$ .

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

Step 6.2

Factor $- 1$ out of $- p^{2}$ .

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

Step 6.3

Factor $- 1$ out of $- (q^{2}) - (p^{2})$ .

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

Step 6.4

Simplify the expression.

Tap for more steps...

Step 6.4.1

Rewrite $- (q^{2} + p^{2})$ as $- 1 (q^{2} + p^{2})$ .

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

Step 6.4.2

Move the negative in front of the fraction.

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