Final answer:
The expression (x^2 - 36)/(42x - 7x^2) simplifies to (x + 6)/7x. The excluded values are x = 0 and x = 6, as these would make the denominator zero.
Step-by-step explanation:
To simplify the rational expression (x2 - 36)/(42x - 7x2), we first factor both the numerator and the denominator. The numerator factors into (x - 6)(x + 6) because it is a difference of squares. The denominator can be factored by taking out a common factor of 7x, resulting in 7x(6 - x).
The simplified form of the expression is obtained by canceling out the common factors in the numerator and denominator.
(x - 6)(x + 6) / 7x(6 - x) = (x + 6) / 7x since (x - 6) and (6 - x) are the same term but with opposite signs, they cancel each other out.
Therefore, the simplified expression is (x + 6)/7x. The excluded values are x = 0 and x = 6, since setting x to either of these values would result in the denominator being zero, which is undefined in mathematics.
Always check the answer to ensure it is reasonable and no terms have been overlooked during the simplification process.