Rationality

Probability Level 3

Find the number of rational terms in the expansion of ( 5 1 6 + 2 1 8 ) 100 \large \left(5^{\frac{1}{6}} + 2^{\frac{1}{8}} \right)^{100}


The answer is 4.

Aniket Sanghi by Aniket Sanghi , 21, India

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

1 solution

Kay Xspre
Jan 31, 2016

For 0 ≤ x ≤ 100 0 ≤ x≤ 100 , each term of this expansion will be in the form of

( 100 x ) 5 x 6 8 100 − x 8 \binom{100}{x}5^{\frac{x}{6}}8^{\frac{100-x}{8}}

Upon verification, there will be 4 terms of ( x , 100 − x ) (x, 100-x) that expand into integer, which is ( 12 , 88 ) , ( 36 , 64 ) , ( 60 , 40 ) , ( 84 , 16 ) (12, 88), (36, 64), (60, 40), (84, 16) , hence the integer terms is 4 terms

0 Helpful 0 Interesting 0 Brilliant 0 Confused

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...