Chapter SR12, Problem 15P

Solution

The least common denominator of the rational expressions $\frac{1}{x-4}\text{ and }\frac{1}{{\left(x-4\right)}^2}$ .

  ${\left(x-4\right)}^2$

Given:

The rational expressions, $\frac{1}{x-4}\text{ and }\frac{1}{{\left(x-4\right)}^2}$ .

Concept Used:

• The opposite operation of addition is subtraction and the opposite operation of subtraction is addition.
• The opposite operation of multiplication is division and the opposite operation of division is multiplication.
• Least Common Denominator (LCD) is the smallest number that is divisible by the denominator of all the fractions that are added or subtracted.

Calculation:

In order to find the least common denominator (LCD) for the given rational expressions, find the factors of both denominators and find the product of all the distinct factors and common factors as shown below

  $\begin{array}{c}\text{Factors of denominaotrs:}\\ x-4=x-4\\ {\left(x-4\right)}^2=\left(x-4\right)\cdot \left(x-4\right)\\ \text{Product of factors}=\left(x-4\right)\cdot \left(x-4\right)\\ ={\left(x-4\right)}^2\end{array}$

Thus, the least common denominator of the rational expressions is ${\left(x-4\right)}^2$ .