Questions: Find the least common denominator for the following pair of rational expressions 7/(k^2+6k) and 6/(k^2+2k-24)

Solution Steps

To find the least common denominator (LCD) of two rational expressions, we need to factor the denominators of each expression and then determine the least common multiple (LCM) of these factors. The LCD will be the product of the highest powers of all factors that appear in the factorizations.

We start by factoring the denominators of the given rational expressions. The first denominator is:

\[ k^2 + 6k = k(k + 6) \]

The second denominator is:

\[ k^2 + 2k - 24 = (k - 4)(k + 6) \]

From the factorizations, we have the following factors:

• For \( \frac{7}{k^2 + 6k} \): \( k \) and \( k + 6 \)
• For \( \frac{6}{k^2 + 2k - 24} \): \( k - 4 \) and \( k + 6 \)

To find the least common denominator (LCD), we take the highest power of each factor that appears in the factorizations:

• The factor \( k \) appears in the first expression.
• The factor \( k + 6 \) appears in both expressions.
• The factor \( k - 4 \) appears in the second expression.

Thus, the LCD is:

\[ \text{LCD} = k(k + 6)(k - 4) \]

Expanding the expression for the LCD gives:

\[ \text{LCD} = k(k + 6)(k - 4) = k(k^2 + 2k - 24) = k^3 + 2k^2 - 24k \]

Final Answer