Simplify (p-q)/(p^3q^2)-(p+q)/(p^2q^3)

Solution:

Step 1:

To find a common denominator for $\frac{p - q}{p^3 q^2}$, multiply by $\frac{q}{q}$ to get $\frac{(p - q)q}{p^3 q^3} - \frac{p + q}{p^2 q^3}$.

Step 2:

For $\frac{p + q}{p^2 q^3}$ to have the same denominator, multiply by $\frac{p}{p}$ to obtain $\frac{(p - q)q}{p^3 q^3} - \frac{(p + q)p}{p^3 q^3}$.

Step 3:

Express both fractions with a common denominator of $p^3 q^3$ by multiplying each by an appropriate form of 1.

Step 3.1:

Multiply $\frac{p - q}{p^3 q^2}$ by $\frac{q}{q}$ to get $\frac{(p - q)q}{p^3 q^3} - \frac{p + q}{p^2 q^3} \cdot \frac{p}{p}$.

Step 3.2:

Raise $q$ to the power of 1 to maintain the same base.

Step 3.3:

Combine exponents using the power rule $a^m a^n = a^{m + n}$ to get $\frac{(p - q)q}{p^3 q^{1 + 2}} - \frac{p + q}{p^2 q^3} \cdot \frac{p}{p}$.

Step 3.4:

Add the exponents 1 and 2 to get $\frac{(p - q)q}{p^3 q^3} - \frac{p + q}{p^2 q^3} \cdot \frac{p}{p}$.

Step 3.5:

Multiply $\frac{p + q}{p^2 q^3}$ by $\frac{p}{p}$ to get $\frac{(p - q)q}{p^3 q^3} - \frac{(p + q)p}{p^3 q^3}$.

Step 3.6:

Raise $p$ to the power of 1.

Step 3.7:

Use the power rule to combine exponents and get $\frac{(p - q)q}{p^3 q^3} - \frac{(p + q)p}{p^{1 + 2} q^3}$.

Step 3.8:

Add the exponents 1 and 2 to get $\frac{(p - q)q}{p^3 q^3} - \frac{(p + q)p}{p^3 q^3}$.

Step 4:

Combine the numerators over the common denominator to obtain $\frac{(p - q)q - (p + q)p}{p^3 q^3}$.

Step 5:

Simplify the numerator.

Step 5.1:

Apply the distributive property to get $\frac{pq - q^2 - (p + q)p}{p^3 q^3}$.

Step 5.2:

Multiply $q$ by $q$.

Step 5.2.1:

Rearrange to $\frac{pq - q \cdot q - (p + q)p}{p^3 q^3}$.

Step 5.2.2:

Multiply $q$ by $q$ to get $\frac{pq - q^2 - (p + q)p}{p^3 q^3}$.

Step 5.3:

Apply the distributive property again.

Step 5.4:

Distribute to get $\frac{pq - q^2 - p \cdot p - qp}{p^3 q^3}$.

Step 5.5:

Multiply $p$ by $p$.

Step 5.5.1:

Rearrange to $\frac{pq - q^2 - p \cdot p - qp}{p^3 q^3}$.

Step 5.5.2:

Multiply $p$ by $p$ to get $\frac{pq - q^2 - p^2 - qp}{p^3 q^3}$.

Step 5.6:

Subtract $qp$ from $pq$.

Step 5.6.1:

Rearrange to $\frac{-q^2 - p^2 + pq - qp}{p^3 q^3}$.

Step 5.6.2:

Subtract $pq$ from $pq$ to get $\frac{-q^2 - p^2}{p^3 q^3}$.

Step 5.7:

Combine $-q^2 - p^2$ and $0$ to get $\frac{-q^2 - p^2}{p^3 q^3}$.

Step 6:

Factor out common terms.

Step 6.1:

Factor $-1$ out of $-q^2$ to get $\frac{-1(q^2) - p^2}{p^3 q^3}$.

Step 6.2:

Factor $-1$ out of $-p^2$ to obtain $\frac{-1(q^2) - 1(p^2)}{p^3 q^3}$.

Step 6.3:

Factor $-1$ out of the entire numerator to get $\frac{-1(q^2 + p^2)}{p^3 q^3}$.

Step 6.4:

Simplify the expression.

Step 6.4.1:

Rewrite as $-1(q^2 + p^2)$ to get $\frac{-1(q^2 + p^2)}{p^3 q^3}$.

Step 6.4.2:

Move the negative in front of the fraction to get $-\frac{q^2 + p^2}{p^3 q^3}$.

Knowledge Notes:

1. Common Denominator: To combine fractions, they must have the same denominator. This often involves multiplying by a form of 1 (e.g., $\frac{p}{p}$ or $\frac{q}{q}$) to achieve a common denominator without changing the value of the fractions.

2. Power Rule: When multiplying like bases, the exponents are added together (e.g., $a^m \cdot a^n = a^{m+n}$).

3. Distributive Property: This property allows us to multiply a single term by each term within a parenthesis (e.g., $a(b + c) = ab + ac$).

4. Simplifying Expressions: Involves combining like terms, factoring out common factors, and reducing fractions to their simplest form.

5. Negative Exponents and Factoring: Factoring out a negative can help simplify expressions, especially when dealing with subtraction in numerators or denominators.