simpliify this equation fully?

(x^2 - 10x + 25)/(x^2 - 25)
= (x^2 - 5x - 5x + 25)/(x^2 + 5x - 5x - 25)
= (x - 5)(x - 5)/(x + 5)(x - 5) (cancel out x - 5)
= (x - 5)/(x + 5)

(x - 5) (x - 5)
-----------------------
(x - 5) (x + 5)

(x - 5) / (x + 5)

what does ^ mean?

Factor the numerator: x^2 - 10x + 25 = (x - 5)(x - 5)
Factor the denominator: x^2 - 25 = (x + 5)(x - 5)

The numerator is perfect-square trinomial. You should learn the pattern. The denominator is the difference of two squares. Ditto.

Common factors in numerator and denominator ?
Answer: (x - 5)/(x + 5)

x^2 - 10x + 25
--------------------------
x^2 - 25

x^2 - 10x + 25 becomes (x-5)(x-5)
x^2 - 25 becomes (x-5)(x+5)
So, the (x-5) cancel out.
x^2 - 10x + 25
--------------------------
x^2 - 25
( x- 5 )
= -------
( x + 5 )

= (x-5)^2/[(x-5)(x+5)]
= (x-5)/(x+5)

for x^2 - 10x +25,

simplified it will give you (x -5)^2

as for x^2 - 25,
simplified, you'll get (x+5)(x-5)

hope this helps!(: