The value of (x^a / x^b)^(a+b) * (x^b / x^c)^(b+c) * (x^c / x^a)^(c+a) is?

Appeared in: DSSSB October 2024

Explanation

The expression is simplified by applying the laws of exponents in sequence: quotient rule, power rule, and then product rule.
The quotient rule (xᵃ / xᵇ = xᵃ⁻ᵇ) is applied to each term within the parentheses.
The power rule ((xᵐ)ⁿ = xᵐⁿ) combined with the algebraic identity (a-b)(a+b) = a² - b² simplifies each term to xᵃ²⁻ᵇ², xᵇ²⁻ᶜ², and xᶜ²⁻ᵃ².
The product rule (xᵐ ⋅ xⁿ = xᵐ⁺ⁿ) is used to combine the terms by adding their exponents.
The sum of the exponents (a² - b² + b² - c² + c² - a²) cancels out completely, resulting in 0.
According to the zero exponent rule, any non-zero base 'x' raised to the power of 0 is equal to 1.

Option A: The result is 1, not 0. An expression equals 0 typically if a factor is 0. According to the zero exponent rule, x⁰ = 1, not 0 (assuming x is not 0).
Option C: The result would be 'x' only if the final simplified exponent was 1 (i.e., x¹). In this problem, the exponent simplifies to 0.
Option D: The result is positive 1. A negative result is not possible through these standard exponentiation rules unless the base was negative and raised to an odd power, which is not the case here.

Clinical Relevance

Nursing practice connection: This is primarily an exam-oriented knowledge point with limited direct bedside application, so retain Simplification of algebraic expressions using the laws of exponents as background academic context rather than a clinical decision trigger.
This question tests foundational mathematical aptitude, a skill required for various calculations in nursing, such as drug dosage, IV drip rates, and interpreting data.
What if? If the final exponent had simplified to 1 instead of 0, the correct answer would have been 'x'.

How to Approach the Question

First, identify that the problem involves simplifying an expression with exponents.
Look at the structure: there are three terms being multiplied, and each term has a fraction raised to a power.
Simplify inside the parentheses first by applying the quotient rule for exponents: xᵃ / xᵇ = xᵃ⁻ᵇ.
Next, apply the power rule for exponents: (xᵐ)ⁿ = xᵐⁿ. Recognize the pattern (a-b)(a+b) to simplify the multiplication to a² - b².
Combine the three simplified terms using the product rule: xᵐ ⋅ xⁿ = xᵐ⁺ⁿ. This means adding all the exponents together.
Calculate the sum of the exponents and observe that they cancel out to zero.

Concept Tested & Keywords

Concept Tested: Simplification of algebraic expressions using the laws of exponents.
Stem keywords: exponents, algebra, simplification, value of
Lead-in keywords: value of

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Practise the full DSSSB October 2024

Attempt every question from this paper in a timed mock, then review the full solution for each one.