If $$ \left(\frac{a}{b}\right)^{x-1} = \left(\frac{b}{a}\right)^{x-3} $$, then the value of $x$ is

Concept & Formula This problem tests the inversion rule for exponents, which allows us to manipulate fractions to match their bases on both sides of an equation. $$ \left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n $$ If $$ m^x = m^y $$, then $$ x = y $$. Step-by-Step Solution * The given equation has bases that are reciprocals of each other: $$ \frac{a}{b} $$ and $$ \frac{b}{a} $$. * We need to make the bases identical. Let's invert the base on the right side of the equation. * According to the inversion rule, flipping the fraction means we must multiply the entire exponent by $$-1$$: $$ \left(\frac{b}{a}\right)^{x-3} = \left(\frac{a}{b}\right)^{-(x-3)} = \left(\frac{a}{b}\right)^{-x+3} $$ * Now, rewrite the original equation using this new equivalent term: $$ \left(\frac{a}{b}\right)^{x-1} = \left(\frac{a}{b}\right)^{-x+3} $$ * Since the bases are now exactly the same ($$ \frac{a}{b} $$), we can equate their exponents directly: $$ x - 1 = -x + 3 $$ * Solve the resulting linear equation for $x$. Add $x$ to both sides: $$ 2x - 1 = 3 $$ * Add 1 to both sides: $$ 2x = 4 $$ * Divide by 2: $$ x = 2 $$ Exam Strategy & Shortcut Whenever you see an equation with fractional bases flipped (like $$a/b$$ and $$b/a$$), immediately negate one of the exponents and set them equal to each other. You don't need to write out the base change steps. Simply write: $$ (x - 1) = -(x - 3) $$. This expands to $$ x - 1 = -x + 3 $$, leading to $$ 2x = 4 $$, and finally $$ x = 2 $$. This reduces a 1-minute algebraic process into a 10-second mental calculation. Common Pitfall A common error is failing to distribute the negative sign across the entire binomial exponent when inverting the fraction. Students often incorrectly rewrite the right-side exponent as $$-x - 3$$ instead of $$-x + 3$$, which fundamentally ruins the final equation and leads to incorrect answers. Always use parentheses when negating. Final Answer **Therefore, the correct answer is 2.**