Calculating E = 27^(-3^(-1)) A Step-by-Step Guide

by Sebastian Müller 50 views

Hey guys! Let's dive into a fun math problem today. We're going to tackle an expression that might look a little intimidating at first glance, but don't worry, we'll break it down step by step. Our mission is to calculate the value of E, where E = 27(-3(-1)). We have a few options to choose from: A) 1/9, B) -1/3, C) 1/3, D) -9, and E) 3. Buckle up, because we're about to embark on a mathematical adventure!

Understanding the Expression

Before we jump into solving, let's make sure we understand what the expression 27(-3(-1)) really means. The key here is the nested exponents. We have 27 raised to the power of -3, which in turn is raised to the power of -1. Remember, exponents tell us how many times to multiply a number by itself. A negative exponent indicates a reciprocal. So, x^(-1) is the same as 1/x. This concept is crucial for unraveling our problem. We need to tackle the innermost exponent first and then work our way outwards. This approach will help us simplify the expression and avoid confusion. Think of it like peeling an onion – we start with the outer layers and gradually work towards the center.

The expression 27(-3(-1)) involves nested exponents, which means we need to be extra careful about the order of operations. Exponents tell us how many times to multiply a number by itself, and negative exponents introduce the concept of reciprocals. A negative exponent essentially flips the base to the denominator (or vice versa if it's already in the denominator). For instance, x^(-1) is equivalent to 1/x. This property is vital for simplifying our expression. The innermost exponent is -1 applied to 3, which means we need to find the reciprocal of 3. This reciprocal will then become the exponent for 27. By understanding these fundamental concepts, we can approach the problem with a clear strategy. It's like having a roadmap before embarking on a journey – it helps us stay on track and avoid getting lost. So, let's keep these principles in mind as we move forward.

To really get a handle on this, let's think about some simpler examples first. What if we just had 2^(-1)? That's easy, it's just 1/2. Now, what if we had (2(-1))(-1)? That's (1/2)^(-1), which means we take the reciprocal again, and we get 2. See how the negative exponents work? They flip the base. This basic understanding is super important for tackling our main problem. When we see 3^(-1), we should immediately think 1/3. This is the first step in simplifying the exponent of 27. The rest of the problem will fall into place once we've dealt with this initial reciprocal. It's like laying the foundation for a building – if the foundation is solid, the rest of the structure will be stable. So, let's make sure we have a firm grasp on this concept before moving on.

Step-by-Step Solution

Okay, let's get our hands dirty and solve this thing! The first thing we need to do is deal with that innermost exponent: -3^(-1). As we discussed, this means we need to find the reciprocal of 3, which is 1/3. So, we can rewrite the expression as E = 27^(-(1/3)). Now, we have a slightly simpler expression. We've gotten rid of one of the negative exponents, and things are starting to look a bit more manageable. Remember, we're just taking it one step at a time. Each step simplifies the problem a little more, making the final solution easier to reach. It's like climbing a ladder – you don't try to jump to the top, you climb one rung at a time.

Now, we have E = 27^(-(1/3)). This means 27 raised to the power of negative one-third. The negative sign, as we know, means we need to take the reciprocal. And the fractional exponent, 1/3, means we need to find the cube root. So, we're looking for the reciprocal of the cube root of 27. Let's break that down even further. First, what's the cube root of 27? That's the number that, when multiplied by itself three times, gives us 27. If you know your cubes, you'll recognize that 3 * 3 * 3 = 27. So, the cube root of 27 is 3. Now we have the reciprocal of 3, which is simply 1/3. See how we tackled that? We broke it down into smaller, more digestible pieces. This is a great strategy for any math problem – don't try to do everything at once!

To recap, we started with E = 27(-3(-1)), simplified the innermost exponent to get E = 27^(-(1/3)), and then interpreted that as the reciprocal of the cube root of 27. We found the cube root of 27 to be 3, and the reciprocal of 3 is 1/3. Therefore, E = 1/3. We've arrived at our solution! It's like completing a puzzle – each step fits into place, and finally, you see the whole picture. Math problems can be like puzzles, and the feeling of solving one is super satisfying. So, let's double-check our answer against the options we were given.

Checking the Options

Let's look at those options again. We had A) 1/9, B) -1/3, C) 1/3, D) -9, and E) 3. Our calculated answer is 1/3, which perfectly matches option C. Woohoo! We've successfully solved the problem. It's always a good idea to double-check your answer, especially when you have multiple options to choose from. This helps you avoid careless mistakes and ensures you're confident in your solution. Think of it as proofreading your work before submitting it – a little extra effort can make a big difference.

Options A, B, D, and E are incorrect. Option A (1/9) might be a result of incorrectly squaring instead of cubing or messing up the reciprocal. Option B (-1/3) might be a sign error, forgetting that we're dealing with a positive cube root. Option D (-9) seems way off and likely involves a combination of errors. And option E (3) is the cube root of 27, but we needed the reciprocal of that. By eliminating these incorrect options, we can be even more confident that our answer of 1/3 is the right one. This process of elimination is a valuable tool in problem-solving, both in math and in life.

So, there you have it! We've conquered this mathematical challenge. We started with a seemingly complex expression, broke it down into manageable steps, and arrived at the correct answer. Give yourselves a pat on the back! Math can be challenging, but with a systematic approach and a good understanding of the underlying concepts, you can tackle anything. Remember, the key is to take it one step at a time and not get overwhelmed by the big picture. Just like any skill, math improves with practice, so keep at it, guys!

Conclusion

In conclusion, the correct answer to the expression E = 27(-3(-1)) is 1/3, which corresponds to option C. We arrived at this solution by carefully simplifying the nested exponents, understanding the role of negative exponents and fractional exponents, and breaking the problem down into smaller, more manageable steps. This problem highlights the importance of understanding fundamental mathematical concepts and applying them systematically. Remember, guys, math isn't about memorizing formulas; it's about understanding the logic and reasoning behind them. So, keep practicing, keep exploring, and keep challenging yourselves!

I hope you found this explanation helpful and insightful. If you have any more math problems you'd like me to tackle, feel free to share them. Let's keep learning and growing together! Math can be a fascinating journey, and I'm excited to be a part of yours. Keep up the great work, and remember to approach every problem with a positive attitude and a willingness to learn. You've got this! Now go out there and conquer some more math challenges!