Which thing is larger ???

(10)^(1/10) or (2)^(1/3) ????

No calculators allowed !!!

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

CPhill Oct 7, 2014

#1**0 **

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

Guest Oct 7, 2014

#2**+5 **

(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).

geno3141 Oct 7, 2014

#3**+5 **

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

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

This gives

10^{3} ??? 2^{10}

1000 ??? (2^{5})^{2}

1000 ???? (32)^{2}

1000 < 1024

CPhill Oct 7, 2014