How did you learn to simplify radicals? I think I discovered a new, easier method.?

Question

How did you learn to simplify radicals (like simplifying sq.rt.48 to 4 sq.rt.3 or cube root 48 to 2 cube root 6)

I think I discovered a new way to simplify radicals that is simple, easy, works every time, and with slight modification, can work for roots of any power (cube roots, 10th power roots, etc.). I want to see if anyone else has learned how I do it though.

Answer 1

√ 48 = √ (16 x 3) = 4√ 3
48^(1/3) = (8 x 6)^(1/3) = 2 [ 6^(⅓) ]

Answer 2

Factor 48 into primes as
48 = 2 x 2 x 2 x 2 x 3
If sq.rt. is desired, separate pairs of primes as
(2 x 2) and ( 2 x 2). Thus, there are two pairs of 2 which give 2 x 2 = 4 outside the sq.rt. sign and 3 within the sq.rt. sign giving 4 sq.rt. 3
If cu.rt. of 48 is desired, write 48 = (2 x 2 x 2) x 2 x 3
Only one pair of 3 2's is there. So, 2 comes out of cu.rt. and 2 x 3 = 6 remains inside
Thus, cu.rt. of 48 = 2 cu.rt. 6

Answer 3

Factoring is the usual method when not using a calculator.

√48 = √16*3 = √4*4*3 = 4√3

(48)^1/3 = (3*2^3)^1/3 = 2(3)^1/3