What is the easiest way to solve with out a calculator ~ Rank the numbers

A =2^1/ 3 , B =3^1/ 4 and C =5^1/ 6 from smallest to largest

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- Apr 18th 2011, 03:12 PMsundance240Easiest way to solve
What is the easiest way to solve with out a calculator ~ Rank the numbers

A =2^1/ 3 , B =3^1/ 4 and C =5^1/ 6 from smallest to largest - Apr 18th 2011, 03:36 PMProve It
If 2^(1/3) < 3^(1/4), then

(2^(1/3))^3 < (3^(1/4))^3

2 < 3^(3/4)

2^4 < (3^(3/4))^4

16 < 3^3

16 < 27, which is true.

Therefore 2^(1/3) < 3^(1/4)

The actual proof requires you to start with 16 < 27 (which is a known true statement) and get to 2^(1/3) < 3^(1/4), but it's essentially the same, just going in reverse order.

Do the same for the other combinations of numbers you have been given.