Supposedly this expression can be simplified, but I can't seem to isolate a like term?

Question

(x^2-1)^2(x+1)^(1/2)/(x-1)^-(3/2)

The answer is apparently "very easy", with "no tricks" breaking down to a much simpler form with an x^7 somewhere in it. Its been driving me barmy for weeks, as even the on-line calculators blow up trying to reach the answer..I've tried looking for diff's of squares,tried squaring the lot, square rooting it all, but with no denominator it just inflates and complicates

Answer 1

(x^2 - 1)^2(x + 1)^(1/2)/(x - 1)^(-3/2)

First, bring the negative exponent from the bottom up top, switching its sign:

(x^2 - 1)^2(x + 1)^(1/2)(x - 1)^(3/2)

Now factor (x^2 - 1) as a difference of two squares:

((x - 1)(x + 1))^2(x + 1)^(1/2)(x - 1)^(3/2)

Now rewrite ((x - 1)(x + 1))^2 as (x + 1)^2(x - 1)^2:

(x - 1)^2(x + 1)^2(x + 1)^(1/2)(x - 1)^(3/2)

Now combine the exponents with the same base:

(x - 1)^(7/2) * (x + 1)^(5/2).

Now rewrite (x - 1)^(7/2) as ((x - 1)^7)^(1/2), and rewrite (x + 1)^(5/2) as ((x + 1)^5)^(1/2):

((x - 1)^7)^(1/2) * ((x + 1)^5)^(1/2)

Combine under a 1/2 power:

((x -1)^7 * (x + 1)^5)^(1/2)

Now expand:

[(x^7 - 7x^6 + 21x^5 - 35x^4 + 35x^3 - 21x^2 + 7x - 1)* (x^5 + 5x^4 + 10x^3 + 10x^2 + 5x + 1)]^(1/2).

Now multiply out:

(x^12 - 2x^11 - 4x^10 + 10x^9 + 5x^8 - 20x^7 + 20x^5 - 5x^4 - 10x^3 + 4x^2 + 2x - 1)^(1/2).

This is completely "simplified" in the usual sense that everything possible is expanded. But is this really "simpler" than ((x - 1)^7 (x + 1)^5)^(1/2)? You decide.

Answer 2

Expand 1st term & bring last term up:
(x+1)^2 (x-1)^2 (x+1)^(1/2) (x-1)^(3/2) =
(x+1)^(5/2) (x-1)^(7/2)

It's not going to get simpler than that!