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Exponents are a way to represent repeated multiplication of a number. For example, in the equation given, we see the number 3 raised to an exponent. The exponent tells us how many times to multiply the base (3) by itself. So, when you have something like 3^4, it means 3 × 3 × 3 × 3, which equals 81.

Understanding how exponents work is crucial when solving exponential equations. It allows you to recognize patterns and rewrite numbers in a different yet equivalent form. In this exercise, rewriting 81 as 3^4 helps simplify the equation and make it solvable.

Algebraic equations involve variables, numbers, and arithmetic operations. These equations are core to algebra and require finding the value of unknown variables that make the equation true. In the current problem, we are dealing with an equation involving exponents.

When confronted with exponential equations like 3^(2x) = 81, it's often necessary to rewrite numbers to have the same base. This makes it easier to compare the exponents directly. Once you have matching bases, the variables in the exponents can be isolated and solved using basic algebraic principles.
For instance, because 81 can be rewritten as 3^4, our equation simplifies from 3^(2x) = 81 to 3^(2x) = 3^4, making it straightforward to proceed to the next step.

Solving for x is the last and most important step in solving algebraic equations. In this case, once we have the equation 3^(2x) = 3^4, we can equate the exponents since the bases are the same: 2x = 4.

To isolate x, we perform simple algebra by dividing both sides of the equation by 2, yielding x = 4/2, which simplifies to x = 2. This is our final solution.

• First, rewrite the equation with the same base.
• Then, set the exponents equal to each other.
• Lastly, solve for the variable by performing algebraic operations.

Following these steps ensures a systematic approach to solving exponential equations.