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

Mar 22, 2017

See below.

#### Explanation:

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

and making

$\vec{u} = \left(X , Y , Z\right)$
$\vec{v} = \left(X , Z , Y\right)$
$\vec{w} = \left(Y , X , Z\right)$

we have that

${\left\lVert \vec{u} \right\rVert}^{2} \ge \left\langle\vec{v} , \vec{w}\right\rangle$ (Cauchy-Bunyakovsky-Schwarz)

where $\left\langle\cdot , \cdot\right\rangle$ 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 $x = 0$