Question

If$a^x=bc,b^y=ac,c^z=ab,$ then$xyz=$

• A. $0$
• B. $1$
• C. $x+y+z+2$
• D. $x+y+z$

Answer 1

The correct option is C

$x+y+z+2$

Step1. Express the given data:

Since $a^x=bc,b^y=ac,c^z=ab,$

${\left(a^x\right)}^y={\left(bc\right)}^y$

$$\begin{gathered} =b^y{c}^y \\ =ac.c^y \end{gathered}$$

Step2. Find the value of $\mathit{x}\mathit{y}\mathit{z}$:

${\left(a^{xy}\right)}^z={\left(ac\right)}^z.{\left(cy\right)}^z$

$$\begin{gathered} =c^z{a}^z{\left(c^z\right)}^y \\ =\left(ab\right)a^z{\left(ab\right)}^y \\ =\left(ab\right)a^z.a^y{b}^y \\ =\left(ab\right)a^z.a^y.ac \\ =a^{2+z+y}\left(bc\right) \\ =a^{2+z+y}.a^x \\ =a^{2+x+z+y} \end{gathered}$$

$\Rightarrow a^{xyz}=a^{x+y+z+2}$

Comparing the powers on both sides

$\Rightarrow xyz=x+y+z+2$

Hence, the correct option is C.