Decreasing powers
Assume that a, b, c, and d are positive integers such that , and . Determine . Do note that this problem is from the AIME and is not my creation.
The answer is 757.
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1 solution
if a^m=b^n , where a,b are reals, then there exists a real integer x such that (x^n)^m=(x^m)^n.
therefore a^5=b^4 => (x^4)^5=(x^5)^4; lly for c,d => (y^2)^3=(y^3)^2;
a=x^4, c=y^2;
and c-a=19;
the only solution satisfying this pair is (x=3,y=10);
so the values are a=81,b=243,c=100,d=1000; so d-b=757;