Question

If 2^x=4^y=8^z then prove that 1/2x+1/4y+1/8z= 11/96

Answer 1

Step-by-step explanation:

Given,

2^x = 4^y = 8^z

⇒ 2^x = 2^(2y) = 2^(3z)

Compare power with same base 2.

therefore,

x = 2y = 3z ……………(1)

xyz = 288 (Given)

x.(x/2).(x/3) = 288

x³/6 = 288

x³ = 288×6 ⇒ x = (1728)⅓ ⇒ x = 12

then, y = 12/2 ⇒ y = 6

and z = 12/3 ⇒ z = 4

therefore,

(1/2x) + (1/4y) + (1/8z) = [1/(2×12)] + [1/(4×6)] + [1/(8×4)]

(1/2x) + (1/4y) + (1/8z) = (1/24) + (1/24) + (1/32)

(1/2x) + (1/4y) + (1/8z) = (2/24) + (1/32)

(1/2x) + (1/4y) + (1/8z) = (1/12) + (1/32)

(1/2x) + (1/4y) + (1/8z) = 11/(24×4)

(1/2x) + (1/4y) + (1/8z) = 11/96