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4 Answers

I hope the following is what you're looking for:
5√75 = 5√(3*25) = 5√3*√25 = 5*5*√3 = 25*√3
4√50 = 4√(2*25) = 4√2*√25 = 4*5*√2 = 20*√2
√48 = √(4*4*3) = √4*√4*√3 = 4√3
 
Thus: 
5√75 -4√50 + √48 = 25√3 - 20√2 + 4√3
                            = 29√3 - 20√2
 The idea is Use " Fundamental Arithmetic Theorem"
 
   Convert every composite number to the product of its prime factors:
 
      75 = 2 * 52
       √75 = √ ( 2 * 52 ) = √2 * √52 = 5 √2
 
        You can multiply and divide roots( square roots), to have an answer in the form of square roots with
           the product and quotient of the 2 values.
     
        However, only like radicals add up by adding their multipliers.
 
       
 
5√75 -4√50 + √48 = take this equation and break them into perfect square factors.
5√(25*3) -4√(25*2) + √(16*3)  Perfect square can come out see below for the rules
5*5√3 -4*5√2 + 4√3 
25√3 -20√2 + 4√3   Now you can old add / subtract the front value with the same radical value(√)
(25+4)√3 -20√2  Simplify
 
29√3 -20√2  Simplest form
 
 
RULES FOR PERFECT SQUARE
1^2 = 1 & √1 = 1
2^2 = 4 & √4 = 2
3^2 = 9 & √9 = 3
4^2 = 16 & √16 = 4
5^2 = 25 & √25 = 5
6^2 = 36 & √36 = 6
7^2 = 49 & √49 = 7
8^2 = 64 & √64 = 8
9^2 = 81 & √81 = 9
10^2 = 100 & √100 = 10
11^2 = 121 & √121 = 11
12^2= 144 & √144 = 12
13^2 = 169 & √169 = 13
14^2 = 196 & √196 = 14
15^2 = 225 & √225 = 15
20^2 = 400 & √400 = 20