Assume that the expressions given are denominators of fractions. Find the least common denominator (LCD) for the group. \[ 9 c+2 r, 81 c^{2}-4 r^{2}, 9 c-2 r \] The least common denominator is

Step 1 ：Assume that the expressions given are denominators of fractions. We need to find the least common denominator (LCD) for the group: \(9c + 2r\), \(81c^2 - 4r^2\), \(9c - 2r\).

Step 2 ：The least common denominator (LCD) is the least common multiple (LCM) of the denominators. To find the LCD, we need to factorize each denominator and then take the highest power of each factor that appears in any of the denominators.

Step 3 ：The denominators are: \(9c + 2r\), \(81c^2 - 4r^2\), \(9c - 2r\).

Step 4 ：The second denominator is a difference of squares, which can be factored as \((9c - 2r)(9c + 2r)\).

Step 5 ：So, the denominators can be rewritten as: \(9c + 2r\), \((9c - 2r)(9c + 2r)\), \(9c - 2r\).

Step 6 ：The LCD is the product of the highest powers of all factors that appear in any of the denominators. In this case, the factors are \(9c - 2r\) and \(9c + 2r\). Both factors appear to the power of 1 in the denominators, so the LCD is \((9c - 2r)(9c + 2r)\), which simplifies to \(81c^2 - 4r^2\).

Step 7 ：The least common denominator (LCD) is \(-81c^2 + 4r^2\), which is the same as \(81c^2 - 4r^2\) because the sign does not affect the value of the denominator in a fraction. Therefore, the LCD is \(81c^2 - 4r^2\).

Step 8 ：Final Answer: The least common denominator (LCD) is \(\boxed{81c^2 - 4r^2}\).