Final answer:
The expression ((-5/(x² - 1))⁵/₂)/(x - 1) is simplified by factoring x² - 1 into (x - 1)(x + 1), then cancelling out the (x - 1) term, resulting in ((-5/(x + 1))⁵/₂) with the condition x ≠ 1.
Step-by-step explanation:
To simplify the expression ((-5/(x² - 1))⁵/₂)/(x - 1), we first note that x² - 1 is a difference of squares and can be factored into (x + 1)(x - 1). Hence, the given expression can be rewritten as ((-5/(x + 1)(x - 1))⁵/₂)/(x - 1). Now, we have the same (x - 1) term in the denominator of the fraction inside the exponent and the denominator of the entire expression. Since we are dividing by (x - 1), we can simplify the expression by cancelling out one of the (x - 1) terms.
In effect, our expression simplifies to ((-5/(x + 1))⁵/₂), assuming x ≠ 1 to avoid division by zero. It's worth mentioning that since we are dealing with a fractional exponent, the simplification presumes that we are only considering the principal root.
Finally, to simplify the entire expression ((-5/(x² - 1))⁵/2)/(x - 1), we divide the expression under the root sign by (x - 1). So, the simplified expression is: (-5√((-5)^4))/(√(x² - 1))^5/(x - 1).