# DBT JRF 2022 — Question 45

Exam: DBT JRF 2022 · Section: A · Topic: Aptitude › Algebra & Exponents · Difficulty: Medium

## Question

If $x = y^a$ , $y = z^b$ , $z = x^c$ , find the value of abc.

## Options

- **A.** 1
- **B.** 3
- **C.** 0
- **D.** Cannot be determined

## Answer

Option A

## Explanation

Solving $x = y^a$ , $y = z^b$ , $z = x^c$ for abc: Substitute chain: $x = y^a = (z^b)^a = z^{ab}$ $z^{ab} = (x^c)^{ab} = x^{abc}$ But we started with $x = x^{abc}$ , so: $x^{abc} = x^1 \Rightarrow abc = 1$ (This holds for $x \neq 0, 1$ ) Verification: if $x = 4$ , $y = 2$ , $z = \sqrt{2}$ : $a = \log_y x = \log_2 4 = 2$ ; $b = \log_z y = \log_{\sqrt{2}} 2 = 2$ ; $c = \log_x z = \log_4 \sqrt{2} = 1/4$ ; $abc = 2 \times 2 \times \frac{1}{4} = 1$ ✔ Answer: $abc = 1$ (Option 1).

Source: https://mypyqbuddy.com/questions/if-x-y-a-y-z-b-z-6a104cd5920734408bc0c70a