Is the square root of 5 plus the square root 5 equal to the square root of 10?

1 Answer
Sep 9, 2015

Answer:

No:

#sqrt(5)+sqrt(5) = 2sqrt(5) = sqrt(4)sqrt(5) = sqrt(20) ~~ 4.472135955#

#sqrt(10) = sqrt(2*5) = sqrt(2)sqrt(5) ~~ 3.16227766#

Explanation:

In the above answer, I use #sqrt(ab) = sqrt(a)sqrt(b)# - which is true for any #a, b >= 0#.

Generalisation

Suppose #f# is a function that takes any Real number and gives us another Real number.

Under what circumstances would we expect the following:

#f(a+b) = f(a) + f(b)# for all #a# and #b#?

Actually the only kind of functions that behave like this are all linear functions which take the form:

#f(x) = mx + c#, where #m# and #c# are constants.

In addition #f(0) = 0#, so #c = 0# and #f(x)# must take the form:

#f(x) = mx# for some constant #m#.

The function #f(x) = sqrt(x)# is not linear, so does not behave in this way.