Finding The Value

Number Theory Level 1

If $m=a^x$ , $n=a^y$ and $a^2=(m^yn^x)^z$ , find the value of $xyz$ .

1 solution

Ram Mohith
May 16, 2018

Given that, $m = a^x , n = a^y$

$a^2 = (m^yn^x)^z \implies a^2 = ((a^x)^y(a^y)^x)^z$

$a^2 = (a^{xy} \times a^{xy})^z$

$\implies a^2 = (a^{2xy})^z \implies a^2 = a^{2xy \times z}$

$\implies 2 = 2xyz$

$\implies \color{#20A900} xyz = 1$