Evaluating (-2/5)³: A Step-by-Step Guide

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Evaluating (-2/5)³: A Step-by-Step Guide

Hey guys! Today, we're diving into a math problem that might seem a bit daunting at first, but trust me, it's totally manageable once we break it down. We're going to evaluate the expression (-2/5)³. This involves understanding exponents and how they interact with fractions and negative numbers. So, grab your thinking caps, and let's get started!

Understanding Exponents

Before we jump into the specifics of our problem, let's quickly recap what exponents actually mean. An exponent tells you how many times to multiply a number (the base) by itself. For example, if we have x³, that means we multiply x by itself three times: x * x * x. It’s crucial to remember that the exponent applies to everything inside the parentheses. So, in our case, the exponent 3 applies to both the -2 and the 5.

Think of exponents as a shorthand way of writing repeated multiplication. Instead of writing (-2/5) * (-2/5) * (-2/5), which can get quite tedious, especially with larger exponents, we can simply write (-2/5)³. This makes the expression much cleaner and easier to work with. However, it's important to understand the underlying concept to correctly apply the exponent.

Now, let's consider what happens when we have a negative base and a positive exponent. This is where things can get a little tricky. When you raise a negative number to an odd power (like 3 in our case), the result will be negative. This is because a negative times a negative is a positive, but a positive times a negative is a negative. On the other hand, if you raise a negative number to an even power (like 2 or 4), the result will be positive because the negative signs will cancel out in pairs.

For example, (-2)² = (-2) * (-2) = 4, while (-2)³ = (-2) * (-2) * (-2) = -8. Keeping this rule in mind will help you avoid common mistakes when dealing with exponents and negative numbers. Remember, the sign of the result is just as important as the numerical value!

Breaking Down (-2/5)³

Okay, now that we've refreshed our memory on exponents, let's tackle our main problem: evaluating (-2/5)³. This means we need to multiply the fraction -2/5 by itself three times. So, we can write this out as:

(-2/5) * (-2/5) * (-2/5)

To make this easier to visualize, imagine you're stacking three identical blocks, each representing -2/5, on top of each other. The total height of the stack represents the result of this multiplication. Breaking it down visually can sometimes help to understand the process better.

Now, when we multiply fractions, we multiply the numerators (the top numbers) together and the denominators (the bottom numbers) together. This is a fundamental rule of fraction multiplication, and it's essential to remember it. So, let's apply this rule to our expression.

First, we'll multiply the numerators: (-2) * (-2) * (-2). As we discussed earlier, multiplying a negative number by itself an odd number of times will result in a negative number. So, (-2) * (-2) = 4, and then 4 * (-2) = -8. Therefore, the numerator of our final answer will be -8.

Next, we'll multiply the denominators: 5 * 5 * 5. This is a straightforward multiplication: 5 * 5 = 25, and then 25 * 5 = 125. So, the denominator of our final answer will be 125.

Putting it all together, we have -8/125. This is the result of evaluating (-2/5)³.

Step-by-Step Calculation

Let's walk through the calculation step-by-step to make sure everything is crystal clear:

  1. Write out the expression: (-2/5)³ = (-2/5) * (-2/5) * (-2/5)
  2. Multiply the numerators: (-2) * (-2) * (-2) = -8
  3. Multiply the denominators: 5 * 5 * 5 = 125
  4. Combine the results: -8/125

See? It's not so scary when you break it down into smaller steps. Each step is a simple multiplication, and by keeping track of the signs, we can arrive at the correct answer.

The Final Result

So, after all that work, we've found that (-2/5)³ = -8/125. This is our final answer. We've successfully evaluated the expression by understanding the meaning of exponents and applying the rules of fraction multiplication.

It's important to note that -8/125 is a simplified fraction. It cannot be reduced further because 8 and 125 do not share any common factors other than 1. So, we've arrived at the simplest form of our answer.

Common Mistakes to Avoid

Before we wrap up, let's talk about some common mistakes people make when evaluating expressions like this. Being aware of these pitfalls can help you avoid making them yourself.

  • Forgetting the negative sign: As we discussed earlier, the sign is crucial when dealing with negative numbers and exponents. Make sure you keep track of the signs throughout the calculation. A common mistake is to calculate the numerical value correctly but forget to include the negative sign when the result should be negative.
  • Applying the exponent only to the numerator or denominator: Remember, the exponent applies to everything inside the parentheses. So, it's not just -2 cubed or just 5 cubed, but the entire fraction -2/5 cubed. Failing to apply the exponent to both the numerator and denominator will lead to an incorrect answer.
  • Misunderstanding the order of operations: In more complex expressions, it's essential to follow the order of operations (PEMDAS/BODMAS). However, in our case, the only operation is exponentiation, so the order is straightforward. But, if you encounter expressions with multiple operations, always remember to follow the correct order.
  • Not simplifying the final answer: While -8/125 is already in its simplest form, sometimes you might arrive at a fraction that can be further simplified. Always check if the numerator and denominator have any common factors and simplify the fraction if possible.

By keeping these common mistakes in mind, you can significantly improve your accuracy when evaluating expressions with exponents and fractions.

Practice Makes Perfect

Now that we've gone through this example together, the best way to master this concept is to practice! Try evaluating similar expressions with different fractions and exponents. You can even create your own problems to challenge yourself.

For example, you could try evaluating (-1/3)³, (3/4)², or (-2/7)². Working through these problems will solidify your understanding of exponents and fractions. The more you practice, the more comfortable and confident you'll become.

Also, don't hesitate to seek out additional resources if you're still feeling unsure. There are plenty of online tutorials, videos, and practice problems available. Math textbooks and workbooks can also be valuable resources. And, of course, you can always ask your teacher or a classmate for help.

Conclusion

So there you have it! We've successfully evaluated the expression (-2/5)³ and arrived at the answer -8/125. We've also discussed the underlying concepts of exponents, fraction multiplication, and common mistakes to avoid. Hopefully, this step-by-step guide has helped you understand the process and build your confidence in tackling similar problems.

Remember, math is like building blocks. Each concept builds upon the previous one. By mastering the fundamentals, you'll be well-equipped to tackle more complex problems in the future. Keep practicing, keep asking questions, and most importantly, keep having fun with math! You got this!