note that our objective when rationalizing the denominator is to get a number in the bottom with a power of 1. so we multiply by the denominator (to an appropriate power) over itself to accomplish this

here's the problem:

keeping every thing in radical signs.

cuberoot(7) / cuberoot(12)

= cuberoot(7) / cuberoot(12) * (cuberoot(12))^2 / (cuberoot(12))^2

= cuberoot(7)(cuberoot(12))^2 / (cuberoot(12))^3

= cuberoot(7)(cuberoot(12^2)) /12 ..........note that i moved the square inside the cuberoot, why can we do this? you'll see it more explicitly in the next method i use

= cuberoot(7*(12^2))/12 ........i combined the cuberoots using a law of surds (and exponents)

= cuberoot(1008)/12

= cuberoot(8*126)/12

= cuberoot(8)*cuberoot(126)/12

= 2*cuberoot(126)/12

= cuberoot(126)/6

second method, changing radical signs to powers:

cuberoot(7) / cuberoot(12)

= 7^(1/3) / 12^(1/3)

= 7^(1/3) / 12^(1/3) * 12^(2/3) / 12^(2/3) .......i multiply by 12^2/3 since i know when i multiply 12^(1/3) by 12^(2/3) i will add the powers. 1/3 + 2/3 = 1, which is what we want

= [7^(1/3) * 12^(2/3)] / [12^(1/3) * 12^(2/3)]

= [7^(1/3) * 12^{2*(1/3)}] / [12^(1/3) * 12^(2/3)] .....now you see why i could move the square in and out, because as powers, we have the multiple 2/3. but 2/3 = 2*(1/3) = (1/3)*2. the 2 gives the square, the 1/3 gives the cuberoot. this means i have the choice of taking the cuberoot first and the square later, or the square first and the cuberoot later

= [7^(1/3) * (12^2)^(1/3)}] / [12^(1/3 + 2/3)]

= [{7*(12^2)}^(1/3)] / [12^(1/3) * 12^(2/3)] .......this is a law of exponents... (x^m)*(y^m) = (xy)^m

= 1008^(1/3) / 12

= (126*8)^(1/3) / 12

= [8^(1/3) * 126^(1/3)]/12

= 2*126^(1/3) / 12

= 126^(1/3) / 12

= cuberoot(126) / 12