Question

Find the product and list all excluded values of (x^(2)-81)/(x+81)*(x^(2)+81x)/(x^(3)-81x).

Answer 1

The product of the given expression is (x^2 + 81x)/(x(x + 81)), and excluded values where the expression is undefined are x = -81 and x = 0.

Step-by-step explanation:

To find the product and list all excluded values of the expression (x2-81)/(x+81)*(x2+81x)/(x3-81x), we must first simplify the expression and then consider the values for which the original expression is undefined.

Let's simplify step by step:

1. Factor the expressions where possible. The numerator x2 - 81 can be factored as (x + 9)(x - 9), and the denominator x3 - 81x can be factored as x(x2 - 81) which further factors into x(x + 9)(x - 9).
2. After factoring, the expression becomes ((x + 9)(x - 9)/(x + 81)) * ((x2 + 81x)/(x(x + 9)(x - 9))). We can cancel out the common factors of (x + 9)(x - 9) in the numerator and the denominator.
3. The simplified expression is: (1/(x + 81)) * ((x2 + 81x)/x). Simplifying this results in (x2 + 81x)/(x(x + 81)).

The excluded values are those for which the original denominators would be zero. These are x = -81 and x = 0, as these would make the original denominators (x + 81) and x in x(x + 9)(x - 9) equal to zero.

Therefore, the product is (x2 + 81x)/(x(x + 81)), and the excluded values are x = -81 and x = 0.