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

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. \[ \frac{4}{45 m^{2}} \text { and } \frac{2}{27 m-45} \]
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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 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 45m245m^2, which is already factored as 45×m245 \times m^2.
  • The second denominator is 27m4527m - 45. Factoring out the greatest common factor, we get 9(3m5)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 4545 and m2m^2.
  • The factors of the second denominator are 99 and 3m53m - 5.

The LCM must include each factor at its highest power:

  • The LCM of the numerical coefficients 4545 and 99 is 4545.
  • The highest power of mm is m2m^2.
  • The factor 3m53m - 5 appears in the second denominator.

Thus, the least common denominator is: 45×m2×(3m5) 45 \times m^2 \times (3m - 5)

Final Answer

The least common denominator for the given rational expressions is: 45m2(3m5) \boxed{45m^2(3m - 5)}

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