Solving 2^(x+2) = 4^2: A Step-by-Step Guide To Finding X

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Hey everyone! Let's dive into solving the exponential equation 2^(x+2) = 4^2. This type of problem might seem intimidating at first, but with a few key concepts and a step-by-step approach, you'll be able to tackle it with confidence. We'll break down each step, explain the reasoning behind it, and make sure you understand the underlying principles. So, grab a pen and paper, and let's get started!

Understanding Exponential Equations

Before we jump into solving the specific equation, let's quickly review what exponential equations are and the properties we'll be using. At its core, an exponential equation is an equation where the variable appears in an exponent. Think of it like this: instead of multiplying a variable by a constant (like in a linear equation), you're raising a constant to a variable power.

The equation 2^(x+2) = 4^2 perfectly illustrates this. The unknown, 'x', is part of the exponent (x+2). The base of the exponent on the left side is 2, and on the right side, it's 4. To solve this, we'll need to manipulate the equation so that we can directly compare the exponents. This often involves making sure both sides of the equation have the same base. One of the most crucial properties we'll use is the property of equality for exponential equations. This property states that if you have two exponential expressions with the same base that are equal, then their exponents must also be equal. Mathematically, this means that if b^m = b^n, then m = n, provided that b is a positive number not equal to 1. This makes sense intuitively: if you raise the same number to two different powers and get the same result, those powers must be the same!

Another important concept is the ability to rewrite numbers as powers of a common base. For example, we know that 4 can be written as 2 squared (2^2). This kind of manipulation is essential in solving many exponential equations because it allows us to apply the property of equality. By expressing both sides of the equation with the same base, we can simplify the problem and focus solely on the exponents. This is the key strategy we'll use to crack the 2^(x+2) = 4^2 equation.

Remember, practice is key! The more you work with exponential equations, the more comfortable you'll become with these properties and techniques. So, keep an open mind, don't be afraid to make mistakes (that's how we learn!), and let's dive into the solution.

Step-by-Step Solution: 2^(x+2) = 4^2

Okay, let's break down the solution to 2^(x+2) = 4^2 step-by-step. Our goal here is to isolate 'x' and find its value. Remember the key strategy we discussed: we want to get both sides of the equation to have the same base. This will allow us to use the property of equality for exponential equations.

Step 1: Rewrite 4 as a power of 2

Notice that 4 can be expressed as 2 squared (2^2). This is a crucial step because it allows us to have the same base (2) on both sides of the equation. So, we rewrite the equation as:

2^(x+2) = (22)2

Now, the right side has a power raised to another power. This is where another important exponent rule comes into play: the power of a power rule. This rule states that when you raise a power to another power, you multiply the exponents. In other words, (am)n = a^(m*n).

Step 2: Apply the power of a power rule

Applying this rule to the right side of our equation, we get:

2^(x+2) = 2^(2*2)

Which simplifies to:

2^(x+2) = 2^4

Look at that! We've successfully transformed the equation so that both sides have the same base (2). This is a major accomplishment, and it sets us up for the next crucial step.

Step 3: Apply the property of equality for exponential equations

Now that we have 2^(x+2) = 2^4, we can use the property of equality for exponential equations. Remember, this property states that if b^m = b^n, then m = n. In our case, b is 2, m is (x+2), and n is 4. So, we can equate the exponents:

x + 2 = 4

We've transformed the exponential equation into a simple linear equation! This is a huge simplification, and solving for 'x' is now straightforward.

Step 4: Solve for x

To isolate 'x', we subtract 2 from both sides of the equation:

x + 2 - 2 = 4 - 2

Which gives us:

x = 2

And there you have it! We've successfully solved the equation. The value of x that satisfies the equation 2^(x+2) = 4^2 is x = 2. Awesome work!

Verification and Why It Matters

Alright guys, we've got our solution: x = 2. But before we pop the champagne, it's always a good idea to verify our answer. Verification is the process of plugging our solution back into the original equation to make sure it holds true. This step is crucial for catching any potential errors we might have made along the way. Think of it as a safety net for your hard work!

So, let's take our x = 2 and substitute it back into the original equation: 2^(x+2) = 4^2.

Substituting x = 2, we get:

2^(2+2) = 4^2

Simplifying the left side:

2^4 = 4^2

Now, let's evaluate both sides. We know that 2^4 means 2 multiplied by itself four times (2 * 2 * 2 * 2), which equals 16. And 4^2 means 4 multiplied by itself (4 * 4), which also equals 16. So we have:

16 = 16

Bingo! The equation holds true. This confirms that our solution, x = 2, is indeed correct. We can now confidently say that we've solved the equation.

You might be thinking, "Why is this verification step so important?" Well, there are several reasons. First, it helps to catch any arithmetic errors we might have made during the solving process. A simple mistake in addition, subtraction, multiplication, or division can lead to a wrong answer. Verification helps us identify and correct these errors.

Second, verification is especially important when dealing with more complex equations, such as those involving radicals or logarithms. These types of equations can sometimes introduce extraneous solutions, which are solutions that we obtain through the solving process but do not actually satisfy the original equation. Verification is the only way to identify and eliminate these extraneous solutions.

Finally, verifying our solutions builds confidence in our problem-solving abilities. When we know we've checked our work and confirmed our answer, we can move on to the next problem with a sense of accomplishment and assurance. So, make verification a habit, guys! It's a small step that can make a big difference in your mathematical journey.

Key Takeaways and Practice Problems

Alright, let's recap what we've learned and solidify our understanding with some practice! Solving the equation 2^(x+2) = 4^2 was a fantastic exercise in applying key concepts of exponential equations. We walked through the process step-by-step, and now it's time to highlight the main takeaways:

  • Rewrite to a Common Base: The cornerstone of solving many exponential equations is to rewrite both sides so they have the same base. This allows you to directly compare the exponents.
  • Power of a Power Rule: Remember the rule (am)n = a^(m*n). It's crucial for simplifying expressions where a power is raised to another power.
  • Property of Equality for Exponential Equations: This is the golden rule! If b^m = b^n, then m = n (provided b is a positive number not equal to 1). This lets you equate the exponents once you have a common base.
  • Verification is Key: Always, always, always verify your solution by plugging it back into the original equation. This catches errors and builds confidence.

Now, to really master these concepts, practice is essential. Here are a few practice problems to try out:

  1. Solve for x: 3^(2x-1) = 27
  2. Solve for y: 5^(y+3) = 125
  3. Solve for z: 2^(3z+1) = 8^2
  4. Solve for a: 4^(a-2) = 16
  5. Solve for b: 9^(b+1) = 3^(2b+3)

Work through these problems, applying the steps we've discussed. Don't just look at the solutions; actively try to solve them yourself. If you get stuck, review the steps we took in solving 2^(x+2) = 4^2. Pay attention to how we rewrote the bases, applied the power of a power rule, and used the property of equality. The more you practice, the more these techniques will become second nature.

Remember, learning math is like learning any other skill. It takes time, effort, and persistence. Don't get discouraged if you don't get it right away. Keep practicing, keep asking questions, and you'll get there! Good luck with the practice problems, and keep exploring the fascinating world of exponential equations!

Conclusion

So, there you have it, guys! We've successfully navigated the world of exponential equations and conquered the challenge of solving 2^(x+2) = 4^2. We've not only found the solution (x = 2) but also delved into the underlying principles and strategies that make these types of problems solvable. We've covered rewriting to a common base, the power of a power rule, the property of equality for exponential equations, and the crucial importance of verification. This journey has equipped you with a solid foundation for tackling more complex exponential equations in the future.

Remember, the key to mastering math is a combination of understanding the concepts and practicing consistently. Don't be afraid to make mistakes; they're valuable learning opportunities. Keep an open mind, stay curious, and never stop exploring the beauty and power of mathematics. The more you practice and apply these concepts, the more comfortable and confident you'll become in your problem-solving abilities. So, go forth and conquer those exponential equations! You've got this!