Unravelling the Mystery: Solving Exponential Equations Step-by-Step 

Exponential equations often present themselves as enigmatic puzzles, challenging our mathematical acumen and problem-solving skills. Today, we're diving deep into the solution of a particularly intriguing equation:9^{2x+1} = {81^{x-2}}/{3^x}.

Join us on this mathematical journey as we unravel the mystery and unveil the step-by-step solution.

Understanding the Equation:

Before we delve into solving the equation, let's decipher its components. At first glance, the equation may seem daunting, but breaking it down reveals its underlying structure.

The equation can be rewritten as 9^{2x+1} =3^4(x-2)/ 3^x, utilizing the property(81 = 3^4).

Step 1: Simplification

To simplify the equation further, we utilize the properties of exponents. -9^{2x+1} can be rewritten as \((3^2)^{2x+1}\), which simplifies to (3^{4x+2}).
- 3^4(x-2)}/{3^x} simplifies to \(3^{4(x-2)-x}\), which further simplifies to(3^{4x-8-x}) or (3^{3x-8}).

Step 2: Equating Exponents

Now that both sides of the equation are in terms of (3^x), we can equate the exponents: [4x + 2 = 3x - 8]

Step 3: Solving for (x)

Let's isolate \\(x) by bringing all the terms involving (x) to one side of the equation:

[4x - 3x = -8 - 2]

[x = -10]

Step 4: Verification

To ensure the validity of our solution, let's substitute (x = -10) back into the original equation:

[9^{2(-10)+1} = {81^{-10-2}}/{3^{-10}}]

[9^{-19} = {81^{-12}}/{3^{-10}}]

[9^{-19} = {(3^4)^{-12}}/{3^{-10}}]

[9^{-19} = {3^{-48}}/{3^{-10}}]

[9^{-19} = 3^{-48-(-10)}]

[9^{-19} = 3^{-38}]

The equation holds true, confirming that (x = -10) is indeed the solution.

Conclusion:

Solving exponential equations requires a systematic approach and a keen understanding of exponent properties. By breaking down the equation and simplifying it step-by-step, we were able to unveil its solution. Remember, practice makes perfect, and with perseverance, you can conquer even the most challenging mathematical puzzles. Stay curious, keep exploring, and let mathematics illuminate your path to discovery.