Solving For X In The Equation 2^(x+2) = 32

Oct 16, 2025 by ADMIN 43 views

Hey guys! Let's dive into solving this exponential equation step-by-step. Understanding how to manipulate exponents and solve for variables is super important in math, and this problem gives us a perfect opportunity to practice. We're given the equation $2^{x+2} = 32$ , and our mission is to find the value of x that makes this equation true. This type of problem often appears in algebra and is crucial for grasping more complex mathematical concepts later on. So, let's break it down and get to the bottom of it!

Understanding Exponential Equations

To effectively solve for x, we need to understand the basics of exponential equations. An exponential equation is simply an equation where the variable appears in the exponent. The key to solving these equations is often to express both sides of the equation with the same base. In our case, the base on the left side is 2, so we need to think about how we can express 32 as a power of 2. This is where our knowledge of exponents comes in handy. Remember, exponents tell us how many times to multiply a number (the base) by itself. For example, $2^3$ means 2 multiplied by itself three times (2 * 2 * 2), which equals 8. Recognizing these powers is essential for simplifying and solving equations like this one. Understanding these fundamentals makes the process less intimidating and more about logical steps.

Expressing 32 as a Power of 2

Now, let's focus on the number 32. Can we write 32 as a power of 2? Think about it: 2 times 2 is 4, times 2 again is 8, times 2 again is 16, and one more time gives us 32. So, we multiplied 2 by itself five times to get 32. In exponential form, this means $32 = 2^5$ . This is a crucial step in solving our equation because it allows us to have the same base on both sides. By rewriting 32 as $2^5$ , we're setting the stage for equating the exponents, which is the key to unlocking the value of x. This little trick of expressing numbers as powers of the same base is something you'll use again and again in various math problems.

Solving the Equation

Alright, we've got $2^{x+2} = 32$ , and we know that $32 = 2^5$ . So, we can rewrite our equation as $2^{x+2} = 2^5$ . Now, this is where the magic happens! If the bases are the same, then the exponents must be equal for the equation to hold true. This means we can simply set the exponents equal to each other: x + 2 = 5. See how we've transformed an exponential equation into a simple linear equation? This is the power of understanding exponential properties. Now, we're just one step away from finding the value of x.

Isolating x

To find x, we need to isolate it on one side of the equation. We have x + 2 = 5. To get x by itself, we subtract 2 from both sides of the equation. This gives us x = 5 - 2, which simplifies to x = 3. And there we have it! We've successfully solved for x. This step of isolating the variable is fundamental in algebra, and it’s something you’ll use constantly. Remember, whatever operation you perform on one side of the equation, you must also perform on the other side to maintain the balance. So, the solution to our equation is x = 3.

The Answer

So, if $2^{x+2} = 32$ , then the value of x is 3. Looking at the options provided, the correct answer is E. 3. We started with a seemingly complex exponential equation, but by understanding the properties of exponents and using a little algebraic manipulation, we were able to find the solution. Remember, practice makes perfect, so the more you work with these types of problems, the more comfortable you'll become with them. Keep up the great work, guys!

Why Other Options Are Incorrect

It's also helpful to understand why the other options are incorrect. This reinforces our understanding of the solution process and helps us avoid common mistakes. Let's quickly look at why options A, B, C, and D are not the correct answer:

A. -2: If we substitute x = -2 into the original equation, we get $2^{-2+2} = 2^0$ , which equals 1, not 32.
B. -3: Substituting x = -3 gives us $2^{-3+2} = 2^{-1}$ , which equals 1/2, not 32.
C. 4: If x = 4, then $2^{4+2} = 2^6$ , which equals 64, not 32.
D. 2: Substituting x = 2 gives us $2^{2+2} = 2^4$ , which equals 16, not 32.

By checking these options, we can clearly see that only x = 3 satisfies the original equation. This process of elimination is a useful strategy in problem-solving, especially in multiple-choice questions.

Tips for Solving Exponential Equations

To wrap things up, let's recap some key tips for solving exponential equations:

Express both sides with the same base: This is often the most crucial step. Look for ways to rewrite the numbers in the equation as powers of the same base.
Equate the exponents: Once the bases are the same, you can set the exponents equal to each other.
Solve the resulting equation: This will often be a linear equation, which you can solve using basic algebraic techniques.
Check your answer: Always substitute your solution back into the original equation to make sure it's correct.

By following these tips, you'll be well-equipped to tackle a wide range of exponential equations. Keep practicing, and you'll become a pro in no time! Remember, math isn't about memorizing formulas; it's about understanding the underlying concepts and applying them logically. You got this!

Conclusion

So, there you have it! We've successfully solved the equation $2^{x+2} = 32$ and found that x = 3. We walked through the steps of understanding exponential equations, expressing numbers as powers of the same base, equating exponents, and isolating the variable. Remember, the key to mastering math is practice and understanding the concepts. Don't be afraid to break down complex problems into smaller, manageable steps. And most importantly, have fun with it! Math can be challenging, but it's also incredibly rewarding when you finally crack a tough problem. Keep exploring, keep learning, and keep practicing, guys! You're doing awesome!