Problem 41 - A Graphical Approach to College Algebra

Question

Solve each equation. $$2^{3-x}=8$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 0\).

Step 1: Recognize the Structure

First, observe that the number 8 on the right side of the equation can be expressed as a power of 2 because it is a power of 2. Knowing that \(8 = 2^3\), we can rewrite the equation.

Step 2: Rewrite the Equation

Replace 8 in the equation with \(2^3\). This gives us the equation \(2^{3-x} = 2^3\).

Step 3: Equalize the Exponents

Since the bases are the same (both are 2), we can set the exponents equal to each other. Therefore, we have: \(3-x = 3\).

Step 4: Solve for x

Now solve the equation \(3-x = 3\). By subtracting 3 from both sides, we get \(-x = 0\). Then, multiply by -1 to solve for \(x\), giving us \(x = 0\).

Key Concepts

Algebra

Algebra is the branch of mathematics that deals with symbols and the rules for manipulating these symbols. In algebra, we use letters to represent quantities that can change, these are known as variables. When solving equations, algebra allows us to systematically manipulate and rearrange terms so that the variable can be isolated.
Algebraic expressions consist of numbers, variables, and arithmetic operations. Equations are statements that show two expressions are equal. An equation can often look complex, but breaking it down into smaller, more manageable steps can help simplify the process of finding the solution.
In our exercise, we're dealing with an exponential equation, which is a type of algebraic equation where the variable appears in the exponent. By understanding the fundamentals of algebra, such as rearranging terms and knowing properties of operations, solving these equations becomes straightforward.

Properties of Exponents

Exponents are a shorthand way of expressing repeated multiplication. For example, to say three times two multiplied together, we write it as \(2^3\), which equals 8. Understanding the properties of exponents is crucial when solving equations like the one in our exercise.
Key properties of exponents include:

• If the bases are the same, you can set the exponents equal to each other, as was done in the given solution where \(2^{3-x} = 2^3\) was simplified by equating the exponents.
• Any number to the power of zero is 1, thus \(a^0 = 1\).
• When multiplying like bases, you add the exponents: \(a^m \cdot a^n = a^{m+n}\).

These rules help to simplify and solve complex-looking exponential equations by breaking them down into simpler parts. Once simplified, they're often easy to solve using fundamental algebraic principles.

Equation Solving

Solving equations involves finding the value of the unknown variable that makes the equation true. The exercise provided is an excellent example of how the process of equation solving can simplify into straightforward steps when approached correctly.

We started by expressing both sides of the equation in terms of the same base. Recognizing 8 as \(2^3\), we rewrote the equation using like bases. When the bases are consistent, solving becomes a matter of setting the exponents equal to one another.

In the equation \(3-x = 3\), the goal is to isolate the variable \(x\). By performing algebraic steps—subtracting, adding, multiplying, or dividing both sides of the equation by the same number—we systematically solve for \(x\). Here, subtracting 3 from both sides, and then multiplying by -1 yields \(x = 0\). This process showcases the power of combining algebraic manipulation with exponent rules to effortlessly solve exponential equations.