91Ó°ÊÓ

Exponentiation is a fundamental concept in mathematics, represented by the notation \(a^b\), where \(a\) is called the base and \(b\) is the exponent. This operation means \(a\) is multiplied by itself \(b\) times. For example, \(2^3\) means \(2 \times 2 \times 2 = 8\).
Understanding exponentiation is essential when solving equations that involve exponential expressions. Let's break down the given problem: we need to solve \(2^{x-1} = 8\).
First, note that the equation involves the base of 2 and an exponent of \(x-1\). The right-hand side of the equation (8) needs to be expressed with the same base to simplify the equation further.
This process leads us to the next fundamental idea.

Rewriting expressions is a useful strategy to make equations easier to solve. It often involves changing how a number or an expression is presented without changing its value. For the equation \(2^{x-1} = 8\), we rewrite 8 as a power of 2. We know that \(8 = 2^3\).
By rewriting 8 as \(2^3\), the equation now looks like this: \(2^{x-1} = 2^3\).
When the bases of the exponents are the same, we can simply equate the exponents. This simplification is possible because if \(a^c = a^d\), then \(c = d\).
This step transforms the problem into a much simpler algebraic equation, paving the way for the final solution step.

Algebraic equations are equations that contain variables and constants. These equations can often be solved through basic algebraic techniques. In our case, we now have the simpler equation from the exponent form: \(x - 1 = 3\).
To solve this, we isolate the variable \(x\). This is done by performing inverse operations. Here, you add 1 to both sides to get:
\[x - 1 + 1 = 3 + 1\]
which simplifies to:
\[ x = 4 \]
This shows that the solution to the equation \(2^{x-1} = 8\) is \(x = 4\).
Understanding how to rewrite expressions and solve algebraic equations is crucial for tackling such problems effectively. By systematically applying these steps, even seemingly complex problems can be broken down and solved with ease.