Here's a Fun One to Play Around With.....

It appears that you went from Philadelphia to New York City by using Amtrak's Acela while I went by way of Hong Kong -- could you use some frequent flyer miles?

Which thing is larger ???

(10)^(1/10)   or  (2)^(1/3)   ????

No calculators allowed !!!

Show your "workings".......

Didn't solve it!

Tried with:

10^(1/10) = ¹⁰√10  = ¹⁰√(2*5) = (¹⁰√2)(¹⁰√5) = (^(5*2)√2)(^(5*2)√5)= [ √(⁵√2) ][ √(⁵√5) ] 

2^(1/3) = ³√2

(10)^(1/10)    or    (2)^(1/3)  ?

(10)^(1/10)  =  (10)^(3/30)  =  (2)^(3/30) x (5)^(3/30)

(2)^(1/3)  =  (2)^(10/30)  =  (2)^(3/30) x (2)^(7/30)

Remove the common factor of (2)^(3/30), so now the comparison is between  (5)^(3/30)  and  (2)^(7/30).

Since they are both 30th roots, take the 30th power of each and compare  (5)^(3)  and  (2)^(7).

5^3  =  125   and  2^7  128.  (I didn't use a calculator, but I did use my fingers.)

Since  128 > 125,      (2)^(1/3)  is slightly larger than (10)^(1/10).

Good job, geno......!!!!!

Another method is to just raise each side to the 30th power

This gives

10³  ??? 2¹⁰

1000 ???  (2⁵)²

1000 ???? (32)²

1000 < 1024

LOL!!!.......