Simplify the difference quotient

Question

[f(x) − f(a]/[x − a]

if x ≠ a.
f(x) = SQROOT of (x − 27)
(Hint: Rationalize the numerator.)

Answer 1

y(x) = (x-27)^.5

y(a) = (a -27)^.5

y(x) -y(a) = (x-27)^.5 - (a-27)^.5)
then multiply top and bottom by
(x-27)^.5 + (a-27)^.5)
and get
(x-27)-(a-27)/{(x-a)[(x-27)^.5 - (a-27)^.5)]}

= (x-a)/{(x-a)[(x-27)^.5 - (a-27)^.5)]}

= 1/[(x-27)^.5 - (a-27)^.5)]