If $x=y^a, y=z^b, z=x^c$, then the value of $abc$ is

The correct answer is Option a: 1. We are given the system of equations: $$x = y^a \quad \text{--- (Equation 1)}$$ $$y = z^b \quad \text{--- (Equation 2)}$$ $$z = x^c \quad \text{--- (Equation 3)}$$ Let us substitute Equation 2 into Equation 1: $$x = (z^b)^a = z^{ab}$$ Now, substitute Equation 3 into this new expression: $$x = (x^c)^{ab}$$ Using the power of a power rule for exponents ($(x^m)^n = x^{mn}$): $$x^1 = x^{abc}$$ Equating the exponents since the bases are identical: $$abc = 1$$ This matches **Option A**. Note: The textbook answer key incorrectly specifies **Option C** ($3$) as the correct answer. However, our mathematical proof clearly shows that the product of the exponents must be $1$ (Option A). Hence, **Option A** is the correct answer.

Ratio, Proportion, Indices, LogarithmMCQPYQ Jun 23Question 889 of 305

All Questions

If $\displaystyle x=y^a, y=z^b, z=x^c$ , then the value of $\displaystyle abc$ is

Options

For any discrepancies in this question, email contact@cadada.in

Correct Answer

✅ Option a — 1

All Options:

Detailed Solution & Explanation

We are given the system of equations:

x = y^a \quad \text{--- (Equation 1)}

y = z^b \quad \text{--- (Equation 2)}

z = x^c \quad \text{--- (Equation 3)}

Let us substitute Equation 2 into Equation 1:

x = (z^b)^a = z^{ab}

Now, substitute Equation 3 into this new expression:

x = (x^c)^{ab}

Using the power of a power rule for exponents (

\displaystyle (x^m)^n = x^{mn}

x^1 = x^{abc}

Equating the exponents since the bases are identical:

abc = 1

This matches **Option A**.
Note: The textbook answer key incorrectly specifies **Option C** (

\displaystyle 3

) as the correct answer. However, our mathematical proof clearly shows that the product of the exponents must be

\displaystyle 1

(Option A).
Hence, **Option A** is the correct answer.

About This Chapter: Ratio, Proportion, Indices, Logarithm

Paper

Paper 3: Quantitative Aptitude

Weightage

5-7 Marks

Key Topics

Ratio, Proportion, Indices, Logarithms

This chapter covers Ratio, Proportion, Indices, Logarithms and is part of Paper 3: Quantitative Aptitude in the CA Foundation exam.

View Official ICAI Syllabus

Exam Strategy Tip

This topic carries 5-7 Marks weightage. Focus on understanding core concepts rather than memorizing.

Ready to Master Ratio, Proportion, Indices, Logarithm?

Practice all 305 questions with instant feedback, earn XP, track your streaks, and ace your CA Foundation exam.

Start Practicing — It's Free

If x=ya,y=zb,z=xc\displaystyle x=y^a, y=z^b, z=x^cx=ya,y=zb,z=xc, then the value of abc\displaystyle abcabc is