Question

z^(x) = y^(2x) ; 2.4^(x) = 2^(z) ; x + y + z = 16 Find the integral roots and x, y, z.

Answer 1

given ,

z^x = y^2x , 2.4^x = 2^z and x + y + z = 16

2.4^x = 2^z

2.2^2x = 2^z

2^(2x + 1) = 2^z

[ use, P^m = P^n then, m = n]

(2x + 1) = z -------(1)
put z = (2x + 1) in x+ y + z = 16

x + y + 2x + 1 = 16

3x + y = 15 -----(2)

again,

z^x = y^2x
from eqn (1) and (2)

(2x +1)^x = (15 -3x)^2x

this is possible only when ,
x = 0

LHS = (2× 0 +1)^0 = 1
RHS = (15 -3×0)^2×0 = 1

put x = 0 in eqn (1) z = 1
put x = 0 in eqn(2) y = 15

hence, x = 0 , y = 15 and z = 1