Question

find the lcd for the following rational expressions.\\(\frac{7}{10x^5}, \frac{3}{25x^2}\\)\\(\text{lcd} = \square\\) (simplify your answer.)

Explanation:

Step1: Factor the denominators

Factor \(10x^5\): \(10x^5 = 2\times5\times x^5\)
Factor \(25x^2\): \(25x^2 = 5^2\times x^2\)

Step2: Find the LCD of the coefficients

For the coefficients 10 and 25, the prime factors are 2, \(5\), and \(5^2\). The LCD of the coefficients is the product of the highest powers of all prime factors involved, so \(2\times5^2 = 50\).

Step3: Find the LCD of the variable parts

For the variable part \(x\), the highest power of \(x\) in the denominators is \(x^5\) (from \(x^5\) in \(10x^5\)).

Step4: Combine the LCD of coefficients and variable parts

Multiply the LCD of the coefficients (50) by the LCD of the variable parts (\(x^5\)) to get the overall LCD: \(50\times x^5 = 50x^5\)

Answer:

\(50x^5\)