Questions: Find the least common denominator for the following pair of rational expressions. 4/45 m^2 and 2/27 m-45

Solution Steps

To find the least common denominator (LCD) of two rational expressions, we need to determine the least common multiple (LCM) of their denominators. First, factor each denominator completely. Then, identify the highest power of each factor that appears in any of the denominators. The product of these factors will give us the LCD.

First, we factor each denominator of the given rational expressions:

• The first denominator is \(45m^2\), which is already factored as \(45 \times m^2\).
• The second denominator is \(27m - 45\). Factoring out the greatest common factor, we get \(9(3m - 5)\).

To find the least common denominator (LCD), we need the least common multiple (LCM) of the factored denominators:

• The factors of the first denominator are \(45\) and \(m^2\).
• The factors of the second denominator are \(9\) and \(3m - 5\).

The LCM must include each factor at its highest power:

• The LCM of the numerical coefficients \(45\) and \(9\) is \(45\).
• The highest power of \(m\) is \(m^2\).
• The factor \(3m - 5\) appears in the second denominator.

Thus, the least common denominator is: \[ 45 \times m^2 \times (3m - 5) \]

Final Answer