Questions: Find the least common denominator (LCD) of 16x/(x+5) and 9/(5x+25). The least common denominator is

Find the least common denominator (LCD) of 16x/(x+5) and 9/(5x+25). The least common denominator is
Transcript text: Find the least common denominator (LCD) of $\frac{16 x}{x+5}$ and $\frac{9}{5 x+25}$. The least common denominator is $\square$
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Solution

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Solution Steps

To find the least common denominator (LCD) of two rational expressions, we need to identify the denominators of each expression and then determine the least common multiple (LCM) of these denominators. The denominators in this case are \(x+5\) and \(5x+25\). We should first factor these expressions to simplify the process of finding the LCM.

Step 1: Identify the Denominators

The denominators of the given rational expressions are:

  1. \( d_1 = x + 5 \)
  2. \( d_2 = 5x + 25 \)
Step 2: Factor the Denominators

We can factor the second denominator: \[ d_2 = 5(x + 5) \]

Step 3: Find the Least Common Denominator

To find the least common denominator (LCD), we need to determine the least common multiple (LCM) of the two denominators: \[ \text{LCD} = \text{lcm}(d_1, d_2) = \text{lcm}(x + 5, 5(x + 5)) \] Since \(5(x + 5)\) includes \(x + 5\) as a factor, the least common denominator is: \[ \text{LCD} = 5(x + 5) \]

Final Answer

The least common denominator is \(\boxed{5(x + 5)}\).

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