Chapter 11.4, Problem 16P

Does it matter whether you use the LCD first or the GCF first when adding or subtracting a rational expression with different denominators and simplifying? Use an example to justify your claim.

To determine

To state if it does matter whether to use the LCD first or the GCF first while adding or subtracting a rational expression with different denominators.

Answer to Problem 16P

If the denominators have no common factors, the LCD is calculated first by simplifying their product.

If the denominators have common factors, the GCF is calculated first and then the LCD is calculated by dividing their product by their GCF.

Explanation of Solution

Given information:

The given statement is:

“Does it matter whether you use the LCD first or the GCF first while adding or subtracting a rational expression with different denominators and simplifying? Use an example to justify your claim.”

If the denominators have no common factors, the LCD is calculated first by simplifying their product.

If the denominators have common factors, the GCF is calculated first and then the LCD is calculated by dividing their product by their GCF.

In the expression $\frac{1}{2a}+\frac{1}{b}$ the denominators have no common factors, so the LCD is calculated first by multiplying their denominators.

  $\begin{array}{l}\frac{1}{2a}+\frac{1}{b}\\ =\frac{b+2a}{2ab}\left[\text{Write the LCD of the denominators}\right]\end{array}$

The sum is $\frac{b+2a}{2ab}$ .

In the expression $\frac{1}{2a}+\frac{1}{a}$ the denominators have common factors, so the GCF is calculated first and is used to calculate the LCD.

  $\begin{array}{l}\frac{1}{2a}+\frac{1}{a}\\ =\frac{1+2}{2a}\left[\text{ LCD}=\frac{2a^2}{a}=2a\right]\\ =\frac{3}{2a}\end{array}$

The sum is $\frac{3}{2a}$ .