Question

Why are radicals simplified before adding and subtracting? Explain your reasoning by adding sqrt8 and sqrt32 algebraically. Compare the process to multiplying and dividing.

Answer 1

The first reason is that square root extraction is NOT a linear process, i.e.
√a + √b ≠ √(a+b) unless a = b. What usually happens is that some common factors can be gotten out frpom the radical and then the distributive law allows some consolidation of terms.
sqrt(8) + sqrt(32) = 2sqrt(2) + 4sqrt(2) = 6 sqrt(2)

but sqrt(8) + sqrt(24) = 2sqrt(2) + 2sqrt(6) = 2[sqrt(2) + sqrt(6)].