91Ó°ÊÓ

Algebra is a branch of mathematics that uses symbols and letters to represent numbers and quantities in formulas and equations. In solving exponential equations, algebraic manipulation is crucial. We often use **addition**, **subtraction**, **multiplication**, and **division** to isolate the variable we are solving for.

In the given problem, after simplifying the exponential expressions, you end up with a linear equation:
to solve for \(x\), we isolate it by adding and subtracting terms on both sides.
For example, we had the equation \(2x - 4 = 1\). Adding 4 to both sides gives \(2x = 5\).
Finally, divide both sides by 2 to find \(x = \frac{5}{2}\).

This step-by-step manipulation of terms is a core part of algebra and helps us find the unknown value.

Exponential functions are mathematical functions in which an independent variable is in the exponent. They have the form \(f(x) = b^x\), where \(b\) is the base and \(x\) is the exponent.

In our problem, we deal with the function \(2^{x+1} \times 2^{-3}\). Exponential functions can grow very quickly or decay very quickly, depending on the base and sign of the exponent.

When working with exponential functions, we often use properties of exponents to simplify expressions. One key property is that when you multiply like bases, you add exponents: \(a^m \times a^n = a^{m+n}\).

In the problem, this property helps simplify \(2^{x+1} \times 2^{-3}\) to \(2^{(x+1-3)} = 2^{x-2}\). Understanding these properties is essential in making the solution process smoother.

The power rule in algebra, particularly with exponents, states that \((a^m)^n = a^{mn}\). This is critical when dealing with expressions that already have an exponent and are raised to another power.

Applying this rule transforms complex expressions into simpler forms, as seen in our exercise.

We originally have \((2^{x-2})^2\). By applying the power rule, this becomes \(2^{(x-2) \times 2}\), which simplifies further to \(2^{2x-4}\). Using the power rule can greatly simplify our work with exponential functions.

This step shows how crucial it is to understand and correctly apply the power rule when manipulating exponents, making the process of solving equations more manageable.