Question

If $x=y^{\frac{1}{a}},y=z^{\frac{1}{b}}\text{and}z=x^{\frac{1}{c}}\text{where}x\ne 1,y\ne 1,z\ne 1$, then what is the value of $abc$?

• A. $-1$
• B. $1$
• C. $0$
• D. $3$

Answer 1

Answer: (B)

$1$

Given $x=y^{\frac{1}{a}}.....\left(1\right)$

$y=z^{\frac{1}{b}}.....\left(2\right)$

$z=x^{\frac{1}{c}}.....\left(3\right)$

Putting the value of y from equation (2) in equation (1)

$x=\left[\right(z{)}^{\frac{1}{b}}{]}^{\frac{1}{a}}\Rightarrow x=z^{\frac{1}{ab}}$

Putting the value of z from equation (3) in the above equation

$x=\left[\right(x{)}^{\frac{1}{c}}{]}^{\frac{1}{ab}}\Rightarrow x^1=x^{\frac{1}{abc}}$

$\therefore \frac{1}{abc}=1\Rightarrow abc=1$