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

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Which thing is larger ???

(10)^(1/10)   or  (2)^(1/3)   ????

No calculators allowed !!!   Oct 7, 2014

#4
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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?

Oct 8, 2014

#1
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Which thing is larger ???

(10)^(1/10)   or  (2)^(1/3)   ????

No calculators allowed !!!

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

Oct 7, 2014
#2
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(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).

Oct 7, 2014
#3
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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   Oct 7, 2014
#4
+5

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?

geno3141 Oct 8, 2014
#5
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LOL!!!.......   Oct 8, 2014