Question

If 2xyz = 1 and x ^ 3 + y ^ 3 + z ^ 3 = 1 1. Prove that: a^ *2y-1z-1 * a^ y2.2-1.x-1 * a^ 22.x-1.y-1 =a^ 2

Answer 1

Given : 2xyz = 1  x³ + y³ + z³ = 1  

To Find : Prove that

$a^{x^2y^{-1}z^{-1}}.a^{y^2z^{-1}x^{-1}}.a^{z^2x^{-1}y^{-1}} = a^2$

Solution:

$a^{x^2y^{-1}z^{-1}}.a^{y^2z^{-1}x^{-1}}.a^{z^2x^{-1}y^{-1}} = a^2$

LHS =

$a^{x^2y^{-1}z^{-1}}.a^{y^2z^{-1}x^{-1}}.a^{z^2x^{-1}y^{-1}}$

$=a^{\frac{x^2}{yz}}.a^{\frac{y^2}{xz}}.a^{\frac{z^2}{xy}}$

Using identity  xᵃ . xᵇ = xᵃ⁺ᵇ

$=a^{(\frac{x^2}{yz}+\frac{y^2}{xz}+\frac{z^2}{xy})}$

$=a^{(\frac{x^3+y^3+z^3}{xyz} )}$

x³ + y³ + z³ = 1  

2xyz = 1  

=>   x³ + y³ + z³ = 2xyz

=> (x³ + y³ + z³ )/ xyz = 2

Substituting

$a^{(\frac{x^3+y^3+z^3}{xyz} )}=a^2$

= RHS

Hence proved

$a^{x^2y^{-1}z^{-1}}.a^{y^2z^{-1}x^{-1}}.a^{z^2x^{-1}y^{-1}} = a^2$

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