How you solve this? $4^x+9^x+25^x=6^x+10^x+15^x$

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Answer 1 · 2017-03-22T00:06:38

See below.

Calling $X = 2^x, Y = 3^x, Z = 5^x$ we have

and making

$vec u = (X,Y,Z)$
$vec v = (X,Z,Y)$
$vec w =(Y,X,Z)$

we have that

$norm(vec u)^2 ge << vec v, vec w >>$ (Cauchy-Bunyakovsky-Schwarz)

where $<< cdot, cdot >>$ represents the scalar product of two vectors.

and the equality is verified only for $X=Y=Z$

which implies on

$2^x=3^x=5^x$

This happen for $x=0$

How you solve this? 4^x+9^x+25^x=6^x+10^x+15^x4^x+9^x+25^x=6^x+10^x+15^x