Solving $16^x=64^{x+4}$ A Step-by-Step Guide

by ADMIN 45 views

Hey guys! Today, we're diving into the fascinating world of exponential equations. Exponential equations might seem daunting at first, but with a clear strategy and some fundamental knowledge of exponents, you'll be able to solve them like a pro. In this article, we're going to tackle the equation 16^x = 64^(x+4). We'll break down the problem step-by-step, showing you not only how to arrive at the correct answer but also the underlying principles that make it all click. So, grab your thinking caps, and let's get started!

Understanding Exponential Equations

Before we jump into the solution, let's make sure we're all on the same page about what exponential equations are and the key concepts involved. An exponential equation is an equation in which the variable appears in the exponent. These equations pop up in various fields, from finance (calculating compound interest) to science (modeling population growth and radioactive decay). The beauty of exponential equations lies in their ability to describe phenomena that change rapidly over time.

The core idea behind solving these equations is to manipulate them so that we can compare the exponents directly. This usually involves expressing both sides of the equation with the same base. Remember, a base is the number that is raised to a power (the exponent). For example, in the expression 2^3, 2 is the base and 3 is the exponent. When we have the same base on both sides of the equation, we can equate the exponents and solve for the variable.

To effectively work with exponential equations, it's crucial to have a solid grasp of exponent rules. These rules provide the tools to simplify and rewrite expressions, making it easier to find a common base. Some of the most important exponent rules include:

  • Product of Powers: a^m * a^n = a^(m+n)
  • Quotient of Powers: a^m / a^n = a^(m-n)
  • Power of a Power: (am)n = a^(m*n)
  • Power of a Product: (ab)^n = a^n * b^n
  • Power of a Quotient: (a/b)^n = a^n / b^n
  • Negative Exponent: a^(-n) = 1/a^n
  • Zero Exponent: a^0 = 1 (for a ≠ 0)

Keep these rules handy as we move forward. They'll be our trusty companions as we conquer this equation!

Finding a Common Base

Our equation is 16^x = 64^(x+4). The first thing we need to do is find a common base for both 16 and 64. This means we're looking for a number that, when raised to some power, gives us both 16 and 64. Think about the powers of 2. We know that 2^4 = 16 and 2^6 = 64. Bingo! 2 is our common base. Using a common base is so important for solving exponential equations. It allows us to directly compare the exponents and simplify the equation.

Now, we'll rewrite both sides of the equation using the base 2. We can rewrite 16^x as (24)x and 64^(x+4) as (26)(x+4). See how we're just expressing 16 and 64 in terms of powers of 2? This is a crucial step in solving exponential equations. Without a common base, it's like trying to compare apples and oranges. But with a common base, we're talking the same language, mathematically speaking.

Next, we'll apply the power of a power rule, which states that (am)n = a^(m*n). This rule is super handy because it allows us to simplify expressions with nested exponents. Applying this rule to our equation, we get:

  • (24)x = 2^(4x)
  • (26)(x+4) = 2^(6(x+4))

Our equation now looks like this: 2^(4x) = 2^(6(x+4)). Notice how we've transformed the original equation into a form where both sides have the same base. This is a major breakthrough! We're one step closer to solving for x.

Equating the Exponents

Now that we have the same base on both sides of the equation (2^(4x) = 2^(6(x+4))), we can equate the exponents. This is a direct consequence of the fundamental property of exponential functions: if a^m = a^n, then m = n (provided a is not 0, 1, or -1). In other words, if two exponential expressions with the same base are equal, then their exponents must be equal. This is the magic step that allows us to transform an exponential equation into a more manageable linear equation.

So, we set the exponents equal to each other: 4x = 6(x+4). We've now converted the exponential equation into a linear equation, which is much easier to solve. Linear equations are our old friends from algebra class. We know how to handle these! The key takeaway here is the power of the common base. By expressing both sides of the equation with the same base, we were able to unlock the relationship between the exponents.

Solving the Linear Equation

Our equation is now 4x = 6(x+4). To solve for x, we first need to distribute the 6 on the right side of the equation. This means multiplying 6 by both x and 4 inside the parentheses. This gives us: 4x = 6x + 24. Remember, the distributive property is a fundamental tool in algebra. It allows us to simplify expressions by multiplying a term by each term inside parentheses.

Next, we want to isolate the x terms on one side of the equation. Let's subtract 6x from both sides: 4x - 6x = 6x + 24 - 6x. This simplifies to -2x = 24. By subtracting 6x from both sides, we've effectively moved all the x terms to the left side of the equation. This is a common strategy in solving linear equations.

Now, to solve for x, we need to divide both sides of the equation by -2: (-2x) / -2 = 24 / -2. This gives us x = -12. Ta-da! We've found the value of x that satisfies the equation. Division is the final step in isolating x. We divide by the coefficient of x to get x all by itself on one side of the equation.

Checking the Solution

It's always a good idea to check our solution to make sure it's correct. To do this, we'll plug x = -12 back into the original equation: 16^x = 64^(x+4). Substituting x = -12, we get: 16^(-12) = 64^(-12+4). Now, let's simplify each side.

First, let's rewrite 16^(-12) as (24)(-12) = 2^(-48). Remember, we found earlier that 16 is 2^4. And the power of a power rule tells us that (am)n = a^(m*n).

Next, let's simplify the right side of the equation. We have 64^(-12+4) = 64^(-8). We can rewrite 64 as 2^6, so we have (26)(-8) = 2^(-48). Aha! Both sides are equal: 2^(-48) = 2^(-48). This confirms that our solution, x = -12, is indeed correct. Checking our solution is like the final seal of approval. It gives us confidence that we haven't made any mistakes along the way.

The Answer and Key Takeaways

So, the solution to the equation 16^x = 64^(x+4) is x = -12. That corresponds to answer choice A. Congratulations, guys, we nailed it! Now, let's recap the key steps we took to solve this equation. These steps will serve as a roadmap for tackling other exponential equations in the future.

  1. Find a Common Base: The first crucial step is to express both sides of the equation using the same base. This allows us to compare the exponents directly. In our case, we identified 2 as the common base for 16 and 64.
  2. Apply Exponent Rules: We used the power of a power rule to simplify the expressions. Understanding and applying exponent rules is essential for manipulating exponential equations.
  3. Equate the Exponents: Once we had the same base on both sides, we equated the exponents. This transformed the exponential equation into a linear equation.
  4. Solve the Linear Equation: We used basic algebraic techniques to solve the linear equation for x.
  5. Check the Solution: Finally, we checked our solution by plugging it back into the original equation to ensure it was correct.

By mastering these steps, you'll be well-equipped to tackle a wide range of exponential equations. Keep practicing, and soon you'll be solving them with ease! Exponential equations are a fundamental part of mathematics, and understanding them opens doors to more advanced concepts. So, keep up the great work, and never stop exploring the fascinating world of math!

Practice Makes Perfect

Now that we've walked through this example, the best way to solidify your understanding is to practice. Try solving similar exponential equations on your own. You can find plenty of examples online or in your math textbook. Remember, the more you practice, the more comfortable you'll become with the process.

Try varying the complexity of the equations. Start with simpler ones where the common base is easy to identify, and then gradually move on to more challenging problems. Don't be afraid to make mistakes – they're a natural part of the learning process. The key is to learn from your mistakes and keep pushing forward.

Consider setting up some practice problems for yourself. This is a great way to test your knowledge and identify areas where you might need to focus your efforts. For example, you could try solving equations like 9^x = 27^(x-1) or 25^(x+2) = 125^(2x-1). These equations will give you a good workout and help you develop your problem-solving skills.

And remember, if you get stuck, don't hesitate to seek help. Ask your teacher, a classmate, or consult online resources. There's a wealth of information available, and with a little effort, you can overcome any obstacle. Learning mathematics is like building a house – each concept builds upon the previous one. So, make sure you have a solid foundation in the basics before tackling more advanced topics.

Conclusion

Solving exponential equations is a valuable skill that will serve you well in your mathematical journey. By understanding the underlying principles and practicing regularly, you can master this topic and gain confidence in your abilities. We've covered a lot in this article, from identifying a common base to applying exponent rules and solving linear equations. Remember the key steps, and you'll be well on your way to becoming an exponential equation expert! Keep up the great work, and remember that math can be fun and rewarding. So, embrace the challenge, and enjoy the journey of learning!