Question

If \[ a^x=b^{\frac{1}{y}}=c^{\frac{3}{z}} \] and \[ b^3=ac^2 \] then \(9xy-2zx=\)

• A. \(0\)
• B. \(1\)
• C. \(2\)
• D. \(3\)

Hint:

For equal powers, always assume a common variable. It simplifies exponent problems quickly.

Answer 1

The Correct Option is D

Concept: When multiple powers are equal, let them be equal to a common variable. This helps express all variables in terms of one quantity.

Step 1: Assume common value.
Let: \[ a^x=b^{\frac1y}=c^{\frac3z}=k \] Then: \[ a=k^{\frac1x} \] \[ b=k^y \] \[ c=k^{\frac z3} \]

Step 2: Substitute into given equation.
Given: \[ b^3=ac^2 \] Substitute: \[ (k^y)^3=k^{\frac1x}\cdot\left(k^{\frac z3}\right)^2 \] \[ k^{3y}=k^{\frac1x+\frac{2z}{3}} \] Since bases are equal: \[ 3y=\frac1x+\frac{2z}{3} \]

Step 3: Multiply by \(3x\).
\[ 9xy=3+2zx \] Rearranging: \[ 9xy-2zx=3 \] Thus, the required value is: \[ \boxed{3} \]