Evaluate the following expression:

\( \left( \frac{x_1}{x_2} \right)^2 \cdot \left( \frac{x_2}{x_3} \right)^2 \cdot \left( \frac{x_3}{x_1} \right)^2 =\ ? \)

Concept:

Law of Exponents:

• When expressions with the same base are multiplied, we add their exponents:  \(a^m × a^n = a^{m+n} \)
• When terms form a ratio:  \( \left( \frac{a}{b} \right)^n = \frac{a^n}{b^n} \)
• This question involves simplification using the above laws of exponents.

Calculation:

Given,

Expression =\( ( \left( \frac{x_1}{x_2} \right)^2 × \left( \frac{x_2}{x_3} \right)^2 × \left( \frac{x_3}{x_1} \right)^2 ) \)  

Combine all fractions inside one power:

⇒\( \left( \frac{x_1}{x_2} × \frac{x_2}{x_3} × \frac{x_3}{x_1} \right)^2 \) 

Simplify inside the bracket:

⇒ \( \left( \frac{x_1 × x_2 × x_3}{x_2 × x_3 × x_1} \right)^2 \)  

⇒ 1² = 1 

∴ The correct answer is Option 3