Question

If a,b,c are positive real numbers, then \( \sqrt{a^{-1}b} \times \sqrt{b^{-1}c} \times \sqrt{c^{-1}a} \) is equal to :

• A. abc
• B. \( \sqrt{abc} \)
• C. \( \frac{1}{abc} \)
• D. 1

Hint:

1. Convert negative exponents: \(x^{-1} = 1/x\). So, \( \sqrt{a^{-1}b} = \sqrt{b/a} \), etc. 2. Combine under one square root: \( \sqrt{b/a \cdot c/b \cdot a/c} \) 3. Multiply the fractions inside: \( \sqrt{(bca)/(abc)} \) 4. Cancel terms: Since multiplication is commutative (\(bca = abc\)), the fraction simplifies to 1. \( \sqrt{1} = 1 \).

Answer 1

The Correct Option is D

Concept: This problem involves simplifying an expression with square roots and negative exponents using the laws of exponents. Key laws of exponents:
\(x^{-n} = \frac{1}{x^n}\)
\(\sqrt{x} = x^{1/2}\)
\(\sqrt{xy} = \sqrt{x}\sqrt{y}\) or \(\sqrt{x}\sqrt{y}\sqrt{z} = \sqrt{xyz}\)
\(x^m \times x^n = x^{m+n}\)
\(\frac{x^m}{x^n} = x^{m-n}\)
\(x^0 = 1\) (for \(x \neq 0\)) Step 1: Rewrite terms with negative exponents We are given the expression: \( \sqrt{a^{-1}b} \times \sqrt{b^{-1}c} \times \sqrt{c^{-1}a} \) Using \(x^{-1} = \frac{1}{x}\):
\(a^{-1}b = \frac{1}{a} \cdot b = \frac{b}{a}\)
\(b^{-1}c = \frac{1}{b} \cdot c = \frac{c}{b}\)
\(c^{-1}a = \frac{1}{c} \cdot a = \frac{a}{c}\) So the expression becomes: \[ \sqrt{\frac{b}{a}} \times \sqrt{\frac{c}{b}} \times \sqrt{\frac{a}{c}} \] Step 2: Combine the square roots Using the property \(\sqrt{x}\sqrt{y}\sqrt{z} = \sqrt{xyz}\), we can combine the terms under a single square root: \[ \sqrt{\frac{b}{a} \times \frac{c}{b} \times \frac{a}{c}} \] Step 3: Simplify the expression inside the square root Multiply the fractions inside the square root: \[ \frac{b \times c \times a}{a \times b \times c} \] We can cancel out common terms in the numerator and the denominator:
The 'a' in the numerator cancels with the 'a' in the denominator.
The 'b' in the numerator cancels with the 'b' in the denominator.
The 'c' in the numerator cancels with the 'c' in the denominator. So, the expression inside the square root simplifies to: \[ \frac{abc}{abc} = 1 \] Step 4: Calculate the final value The expression becomes: \[ \sqrt{1} \] And the square root of 1 is 1. \[ \sqrt{1} = 1 \] Therefore, \( \sqrt{a^{-1}b} \times \sqrt{b^{-1}c} \times \sqrt{c^{-1}a} = 1 \).