If $$ \left(\frac{3}{5}\right)^3 \left(\frac{3}{5}\right)^{-6} = \left(\frac{3}{5}\right)^{2x-1} $$, then $x$ is equal to

Concept & Formula This problem tests the product rule of exponents and the property of equating exponents when the bases on both sides of an equation are identical. $$ a^m \times a^n = a^{m+n} $$ If $$ a^m = a^n $$, then $$ m = n $$ (given $$ a \neq 0, 1, -1 $$). Step-by-Step Solution * Observe the left-hand side of the equation. Both terms have the same base: $$ \frac{3}{5} $$. * Apply the product rule by adding the exponents: $$ \left(\frac{3}{5}\right)^{3 + (-6)} = \left(\frac{3}{5}\right)^{-3} $$ * Now, rewrite the full equation with the simplified left side: $$ \left(\frac{3}{5}\right)^{-3} = \left(\frac{3}{5}\right)^{2x-1} $$ * Since the bases are exactly the same on both sides, we can drop the bases and equate the exponents directly: $$ -3 = 2x - 1 $$ * Solve the linear equation for $x$. Add 1 to both sides: $$ -3 + 1 = 2x $$ $$ -2 = 2x $$ * Divide by 2: $$ x = -1 $$ Exam Strategy & Shortcut Since the bases across the entire equation are identical ($$3/5$$), you can bypass writing them down entirely. Jump straight to the exponent arithmetic: $$3 - 6 = 2x - 1$$. That simplifies mentally to $$-3 = 2x - 1$$, meaning $$2x = -2$$, so $$x = -1$$. This takes less than 5 seconds without writing a single fraction. Common Pitfall A common error is mishandling the negative numbers during basic addition and subtraction. Students sometimes calculate $$3 - 6 = 3$$ instead of $$-3$$, which leads to the incorrect equation $$3 = 2x - 1$$, resulting in $$x = 2$$. Pay strict attention to signs during exponent arithmetic. Final Answer **Therefore, the correct answer is -1.**