Solving (4^2)^-5: A Step-by-Step Exponential Guide

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Hey guys! Today, we're going to break down how to solve the expression (42)-5 and express it in exponential form. Don't worry, it's not as intimidating as it looks! We'll go through each step nice and slow, so you can follow along easily. Let's get started!

Understanding the Basics of Exponents

Before diving into the problem, let's brush up on some exponent basics. Exponents, also known as powers, are a way of showing how many times a number (the base) is multiplied by itself. For instance, in the expression a^b, 'a' is the base and 'b' is the exponent or power. This means you multiply 'a' by itself 'b' times. For example, 2^3 = 2 * 2 * 2 = 8. Understanding this fundamental concept is crucial before we tackle more complex problems. Additionally, there are a few key rules of exponents that will be super helpful. One important rule is the power of a power rule, which states that (am)n = a^(m*n). This rule tells us that when you raise a power to another power, you simply multiply the exponents. Another essential rule is dealing with negative exponents. A negative exponent means you take the reciprocal of the base raised to the positive exponent. That is, a^-n = 1/a^n. These rules are like the building blocks that allow us to simplify and solve exponential expressions efficiently. Getting these rules down pat will make solving problems like our (42)-5 a breeze!

Step-by-Step Solution for (42)-5

Okay, let's get right into solving (42)-5. The first thing we want to do is apply the power of a power rule we just talked about. This rule states that (am)n = a^(m*n). Applying this to our problem, we get:

(42)-5 = 4^(2 * -5) = 4^-10

So, we've simplified the expression to 4^-10. Now, remember what we said about negative exponents? A negative exponent means we need to take the reciprocal of the base raised to the positive exponent. Therefore, we can rewrite 4^-10 as:

4^-10 = 1 / 4^10

At this point, we have successfully expressed the original expression in exponential form. While we could calculate 4^10 to get a numerical value, the question specifically asks for the exponential form, so we'll stick with that. However, for those curious, 4^10 = 1,048,576. But remember, the key here is understanding how to manipulate the exponents. To summarize, we started with (42)-5, applied the power of a power rule to get 4^-10, and then dealt with the negative exponent by taking the reciprocal to get 1 / 4^10. That's it! You've solved it.

Alternative Approaches and Simplifications

Now, let's explore some alternative approaches and further simplifications we could make, just to give you a broader perspective. Instead of directly applying the power of a power rule right away, we could first evaluate 4^2. We know that 4^2 = 16. So, we could rewrite the original expression as:

(42)-5 = (16)^-5

Then, dealing with the negative exponent, we would have:

(16)^-5 = 1 / 16^5

Notice that this is equivalent to our previous answer, since 16 is just 4^2. We can further demonstrate this by expressing 16 as 4^2:

1 / 16^5 = 1 / (42)5

Now, applying the power of a power rule again, we get:

1 / (42)5 = 1 / 4^(2 * 5) = 1 / 4^10

This confirms that both approaches lead to the same result. Another interesting simplification involves prime factorization. We know that 4 = 2^2, so we can rewrite our original expression as:

(42)-5 = ((22)2)^-5

Applying the power of a power rule twice, we get:

((22)2)^-5 = (24)-5 = 2^(4 * -5) = 2^-20

Dealing with the negative exponent, we have:

2^-20 = 1 / 2^20

Since 4^10 = (22)10 = 2^20, this result is also consistent with our previous findings. This prime factorization approach highlights how you can manipulate exponential expressions using different bases. By understanding these alternative methods, you gain a deeper understanding of exponential operations and can choose the approach that best suits the problem at hand.

Common Mistakes to Avoid

Alright, let's chat about some common mistakes people often make when dealing with exponents, so you can dodge those pitfalls. One frequent error is messing up the order of operations. Remember, exponents come before multiplication, division, addition, and subtraction. So, in an expression like 2 * 3^2, you need to calculate 3^2 first (which is 9) and then multiply by 2, resulting in 18, not (2 * 3)^2 = 36. Another common mistake is mishandling negative exponents. Remember, a negative exponent doesn't mean the number becomes negative; it means you take the reciprocal. For example, 2^-1 is 1/2, not -2. People often get confused and think that a negative exponent simply makes the base negative, but that's not the case. Also, be careful when dealing with the power of a power rule. Ensure you're multiplying the exponents correctly. For example, (23)2 is 2^(3*2) = 2^6 = 64, not 2^(3+2) = 2^5 = 32. Mixing up multiplication and addition in this context is a common slip-up. Another mistake to watch out for is incorrectly applying the distributive property. The distributive property does not apply to exponents in the same way it does to multiplication over addition. For instance, (a + b)^2 is not equal to a^2 + b^2. Instead, (a + b)^2 = (a + b) * (a + b) = a^2 + 2ab + b^2. Recognizing and avoiding these common mistakes will significantly improve your accuracy and confidence when working with exponents. Keep these points in mind, and you'll be solving exponential problems like a pro in no time!

Practice Problems

To really nail down your understanding of exponential forms, let's run through some practice problems. Working through examples is the best way to solidify your knowledge and build confidence. Here are a few for you to try:

  1. Simplify (53)-2
  2. Express 3^-4 in fractional form
  3. Simplify (2^2 * 23)-1
  4. Evaluate 7^-2 * 7^4
  5. Rewrite (90.5)3 in simplest form

Solutions:

  1. (53)-2 = 5^(3 * -2) = 5^-6 = 1 / 5^6
  2. 3^-4 = 1 / 3^4 = 1 / 81
  3. (2^2 * 23)-1 = (2(2+3))-1 = (25)-1 = 2^-5 = 1 / 2^5 = 1 / 32
  4. 7^-2 * 7^4 = 7^(-2+4) = 7^2 = 49
  5. (90.5)3 = (3)^3 = 27 (Since 9^0.5 is the square root of 9, which is 3)

Take your time to work through these problems, and don't hesitate to refer back to the explanations and rules we discussed earlier. The more you practice, the more comfortable you'll become with manipulating exponents. If you encounter any difficulties, try breaking the problem down into smaller, more manageable steps. And remember, it's perfectly okay to make mistakes – that's how we learn! By actively engaging with these practice problems and reviewing the solutions, you'll reinforce your understanding and develop your problem-solving skills. Keep up the great work!

Conclusion

Alright guys, we've covered a lot in this guide! We started with the basics of exponents, worked through a step-by-step solution for (42)-5, explored alternative approaches, highlighted common mistakes to avoid, and even tackled some practice problems. By now, you should have a solid understanding of how to manipulate exponential expressions and express them in the required form. Remember, the key to mastering exponents is practice, practice, practice! The more you work with these concepts, the more intuitive they will become. So, keep practicing, keep exploring, and don't be afraid to tackle more challenging problems. You've got this! And remember, if you ever get stuck, just revisit this guide or reach out for help. Keep up the awesome work, and happy exponent-solving! You're well on your way to becoming an exponent expert!