Solving $(2^{-3}-2^{-2})^{-1}$: A Step-by-Step Guide

by ADMIN 55 views

Hey guys! Today, we're diving into a fun math problem: finding the result of the operation $(2{-3}-2{-2})^{-1}$. This might look a little intimidating at first, but don't worry, we'll break it down step by step so it's super easy to understand. We'll cover the basics of exponents, fractions, and how to handle negative exponents. So, grab your pencils and let's get started!

Understanding the Problem

Before we jump into solving, let's make sure we understand what the problem is asking. The expression $(2{-3}-2{-2})^{-1}$ involves negative exponents and a fraction, all wrapped up in parentheses with another negative exponent outside. Sounds like a lot, right? But we'll tackle it one piece at a time.

First, let's focus on exponents. Remember that a negative exponent means we're dealing with the reciprocal of the base raised to the positive exponent. For example, $2^{-3}$ is the same as $ rac{1}{2^3}$. This is a key concept to keep in mind as we move forward.

Next, we have a difference inside the parentheses: $(2{-3}-2{-2})$. This means we need to calculate the values of $2^{-3}$ and $2^{-2}$ and then subtract them. Once we have that result, we'll deal with the final exponent of -1.

Finally, that outer $-1$ exponent means we'll take the reciprocal of whatever we get inside the parentheses. So, essentially, we're flipping the fraction. By understanding these individual parts, we can approach the problem with confidence and solve it accurately. Let's get to the nitty-gritty calculations now!

Step-by-Step Solution

Okay, let's break down the solution step-by-step to make sure we don't miss anything. We'll start with the inner part of the expression and work our way outwards. This is a common strategy in math – tackling the innermost operations first.

Step 1: Evaluate $2^{-3}$ and $2^{-2}$

As we discussed earlier, negative exponents indicate reciprocals. So:

  • 2^{-3} = rac{1}{2^3} = rac{1}{2 imes 2 imes 2} = rac{1}{8}

  • 2^{-2} = rac{1}{2^2} = rac{1}{2 imes 2} = rac{1}{4}

See? Not so scary when we break it down. We've now converted the negative exponents into simple fractions. This makes the next step much easier.

Step 2: Calculate $ rac{1}{8} - rac{1}{4}$

Now we need to subtract these fractions. To do this, we need a common denominator. The least common denominator for 8 and 4 is 8. So, we'll rewrite $ rac{1}{4}$ as an equivalent fraction with a denominator of 8:

rac{1}{4} = rac{1 imes 2}{4 imes 2} = rac{2}{8}

Now we can subtract:

rac{1}{8} - rac{2}{8} = rac{1 - 2}{8} = rac{-1}{8}

We've now simplified the expression inside the parentheses to a single fraction: $- rac{1}{8}$. We're getting closer to the final answer!

Step 3: Apply the Outer Exponent: $(-1)$

We're now left with $\left(-\frac{1}{8}\right)^{-1}$. Remember, a negative exponent means we take the reciprocal. So, we need to find the reciprocal of $- rac{1}{8}$.

To find the reciprocal, we simply flip the fraction. The reciprocal of $- rac{1}{8}$ is $- rac{8}{1}$, which is just $-8$.

Final Answer

Therefore, the result of the operation $(2{-3}-2{-2})^{-1}$ is $-8$. Ta-da! We did it!

Why is the Answer -8?

Let's recap why the answer is -8 to make sure everything's crystal clear. We started with the expression $(2{-3}-2-2})^{-1}$. We first dealt with the negative exponents inside the parentheses, converting them into fractions $ rac{1{8}$ and $ rac{1}{4}$. Then, we subtracted these fractions, which required finding a common denominator. This gave us $- rac{1}{8}$. Finally, we applied the outer exponent of -1, which meant taking the reciprocal of $- rac{1}{8}$, resulting in $-8$. Each step built upon the previous one, leading us to the final solution.

Understanding the order of operations and the properties of exponents is crucial in solving these types of problems. It's like building a house – you need a solid foundation before you can add the walls and roof. In this case, the foundation is understanding negative exponents and fractions.

Common Mistakes to Avoid

When tackling problems like this, there are a few common mistakes that students often make. Being aware of these pitfalls can help you avoid them and ensure you get the correct answer. Let's go over a few key ones:

  • Forgetting the Negative Exponent Rule: One of the most common mistakes is forgetting that a negative exponent means taking the reciprocal. For example, some might incorrectly think that $2^{-3}$ is the same as $-\frac{1}{8}$, instead of $\frac{1}{8}$. Always remember to flip the base when you see a negative exponent!
  • Incorrectly Subtracting Fractions: Subtracting fractions requires a common denominator. A common mistake is to subtract the numerators and denominators directly without finding a common denominator first. This will lead to an incorrect result. Remember to find that common denominator before subtracting!
  • Misinterpreting the Order of Operations: Order of operations is super important. We need to handle the parentheses first, then the exponents, and so on. Jumping the gun and applying the outer exponent before simplifying the expression inside the parentheses will lead to a wrong answer.
  • Sign Errors: It’s easy to make mistakes with signs, especially when dealing with negative numbers and fractions. Pay close attention to the signs in each step and double-check your work to avoid these errors.

By being mindful of these common pitfalls, you can significantly improve your accuracy and confidence when solving similar problems.

Practice Problems

Okay, now that we've walked through the solution and discussed common mistakes, let's put your knowledge to the test! Here are a couple of practice problems similar to the one we just solved. Give them a try and see if you can apply the steps we discussed.

  1. (3−2−3−1)−1(3^{-2}-3^{-1})^{-1}

  2. (5−1+5−2)−1(5^{-1}+5^{-2})^{-1}

Take your time, break each problem down step-by-step, and remember the rules we discussed. If you get stuck, go back and review the solution we worked through together. Practice makes perfect, and the more you practice, the more comfortable you'll become with these types of problems.

Conclusion

So, there you have it! We've successfully solved the problem $(2{-3}-2{-2})^{-1}$ and found the answer to be $-8$. We've covered the basics of negative exponents, fractions, and the importance of the order of operations. Remember, breaking down complex problems into smaller, manageable steps is the key to success in math. Don't be afraid to tackle challenging problems – with a little practice and a solid understanding of the fundamentals, you can conquer anything!

I hope this guide has been helpful and has made understanding this problem a little easier. Keep practicing, keep exploring, and keep having fun with math! You guys got this!