Question

If 2^x=25^y=1000^z , then show that z= 2/ 3+6�

Answer 1

Given:

2^x = 25^y = 1000²

To show:

$z = \: \frac{2xy}{3x + 6y}$

Solution:

Consider,

=> 2^x = 25) = 1000* = k(say)

=> 2^x = k | 25^y = k | 1000^z = k

=> 2^x = k | 5^2y = k | 1000^z = k

Take 1000^z = k

=> (5³x2³)^z = k

Squaring on both sides, we get

=> (5³x2³)^2z = k²

=> 5^6zx2^6z = k×k

=> 5^6z x2^6z = 2^x × 5^2y

Equating powers on both sides, we get

x = 6z & 2y = 6z

x = 6z & y = 3z

Now,

$\frac{2xy}{3x + 6y}$

$= > \: \frac{2(6z)(3z)}{3(6z) + 6(3z)}$

$= > \frac{36z {}^{2} }{18z + 18z}$

$= > \frac{36z {}^{2} }{36z}$

$= z$

$= > z = \frac{2xy}{3x + 6y}$

$\huge{ son \: prodigal}$