Final answer:
The expression 1/√(X²-9) simplifies to 1/√(X - 3)√(X + 3), but cannot be directly matched to any of the provided answer choices without additional assumptions about the domain of X.
Step-by-step explanation:
The expression 1/√(X²-9) cannot be simplified directly to any of the given options a) 1/(x-3), b) 1/(x+3), c) 1/√(X+3), or d) 1/√(X-3) without making assumptions about the domain of X. However, if we assume that X > 3 or X < -3 (for the square root to be real and nonzero), we can factore under the square root to obtain (X-3)(X+3). Then we rewrite the original expression using the difference of squares property.
To factore it, we're looking for two numbers that multiply to -9 and add to 0, which are 3 and -3. Using these numbers, we have the factorization:
X² - 9 = (X + 3)(X - 3)
Our expression now becomes:
1/√((X - 3)(X + 3))
The square root of a product is the product of the square roots (assuming both are positive):
1/√(X - 3)√(X + 3)
But since we cannot uniquely determine the simplification to match any of the provided choices without further conditions on X, we must conclude that the expression cannot be directly matched to options a, b, c or d.