Questions: Find the least common denominator for the following pair of rational expressions.
4/45 m^2 and 2/27 m-45
Transcript text: Find the least common denominator for the following pair of rational expressions.
\[
\frac{4}{45 m^{2}} \text { and } \frac{2}{27 m-45}
\]
Solution
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.
Step 1: Factor the Denominators
First, we factor each denominator of the given rational expressions:
The first denominator is 45m2, which is already factored as 45×m2.
The second denominator is 27m−45. Factoring out the greatest common factor, we get 9(3m−5).
Step 2: Determine the Least Common Denominator
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 m2.
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 m2.
The factor 3m−5 appears in the second denominator.
Thus, the least common denominator is:
45×m2×(3m−5)
Final Answer
The least common denominator for the given rational expressions is:
45m2(3m−5)