Evaluating $0.677+(-0.917--0.731-0.814)^3$: A Math Solution

Oct 13, 2025 by ADMIN 60 views

Hey guys! Let's break down this math problem together. We're diving into evaluating the expression $0.677+(-0.917--0.731-0.814)^3$ . Sounds intimidating? Don't worry, we'll take it step by step. We'll start by simplifying inside the parentheses, then tackle the exponent, and finally, we'll add it all up. Grab your calculators (or your mental math muscles) and let's get started!

Understanding the Expression

Before we jump into calculations, let’s make sure we understand what the expression is asking us to do. The expression is:

$0.677+(-0.917--0.731-0.814)^3$

This involves several arithmetic operations: addition, subtraction (including subtracting a negative number), and exponentiation. The order of operations (PEMDAS/BODMAS) will be crucial here. Remember, PEMDAS stands for Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). Let’s keep this in mind as we proceed. Our main keywords here are evaluating expressions, order of operations, and arithmetic. We'll be using these concepts throughout the solution. First, we'll focus on what's inside the parentheses, because that's our first priority according to PEMDAS. Inside the parentheses, we have a series of additions and subtractions involving decimal numbers. We need to be super careful with the signs to make sure we get the correct result. A small error in the beginning can throw off the entire calculation, so let's take our time and double-check each step. The ability to accurately work with decimals and negative numbers is a fundamental skill in mathematics, and mastering it will help us tackle more complex problems down the road. So, let’s roll up our sleeves and dive into the first step!

Step 1: Simplifying Inside the Parentheses

Okay, let’s tackle the stuff inside the parentheses first: $(-0.917 - -0.731 - 0.814)$ . Remember, subtracting a negative number is the same as adding its positive counterpart. So, $(-0.917 - -0.731)$ becomes $(-0.917 + 0.731)$ . This simplifies our expression inside the parentheses to:

$(-0.917 + 0.731 - 0.814)$

Now, let’s add $-0.917$ and $0.731$ . When adding numbers with different signs, we subtract their absolute values and use the sign of the number with the larger absolute value. So, $|-0.917| = 0.917$ and $|0.731| = 0.731$ . Subtracting, we get $0.917 - 0.731 = 0.186$ . Since $-0.917$ has the larger absolute value, our result is negative:

$-0.186$

Now we have:

$(-0.186 - 0.814)$

Next, we subtract $0.814$ from $-0.186$ . Subtracting a positive number from a negative number is the same as adding the absolute values and keeping the negative sign. So, we add $0.186$ and $0.814$ :

$0.186 + 0.814 = 1.000$

Since both numbers were effectively negative, our result is $-1$ . Therefore, the simplified expression inside the parentheses is:

$-1$

Great! We've simplified the parentheses part. This was a crucial step, and we navigated through the addition and subtraction of decimals like pros. Now that we have a single number inside the parentheses, we can move on to the next operation in our PEMDAS journey, which is handling the exponent. This is where things get a little more interesting, so let's jump right in!

Step 2: Handling the Exponent

Now that we've simplified the expression inside the parentheses to $-1$ , we can focus on the exponent. Our expression now looks like this:

$0.677 + (-1)^3$

The exponent here is $3$ , which means we need to raise $-1$ to the power of $3$ . In other words, we need to multiply $-1$ by itself three times:

$(-1)^3 = (-1) imes (-1) imes (-1)$

Let's break it down step by step. First, $(-1) imes (-1)$ equals $1$ because a negative number multiplied by a negative number gives a positive number. So, we have:

$1 imes (-1)$

Now, we multiply $1$ by $-1$ . A positive number multiplied by a negative number gives a negative number, so:

$1 imes (-1) = -1$

Therefore, $(-1)^3 = -1$ . This is a fundamental concept in mathematics: any negative number raised to an odd power will result in a negative number. Our expression now simplifies to:

$0.677 + (-1)$

We’re almost there! We’ve handled the parentheses and the exponent. Now all that’s left is one simple addition. This is the final stretch, and we're going to nail it. Let's move on to the last step and bring this problem to a satisfying conclusion!

Step 3: Final Addition

We've made it to the final step! Our expression is now:

$0.677 + (-1)$

Adding a negative number is the same as subtracting its absolute value. So, we can rewrite this as:

$0.677 - 1$

Now, we're subtracting a larger number ( $1$ ) from a smaller number ( $0.677$ ). This means our result will be negative. To find the difference, we subtract $0.677$ from $1$ :

$1 - 0.677$

To do this, we can align the decimal points and perform the subtraction:

1. 677

We need to borrow from the left:

99(10)

1. 677

Subtracting, we get:

Since we were subtracting from a larger number and the original expression was $0.677 - 1$ , our final result is negative:

$-0.323$

So, the final answer to the expression $0.677 + (-0.917 - -0.731 - 0.814)^3$ is $-0.323$ . We did it! We successfully evaluated the expression by carefully following the order of operations and handling decimals and negative numbers with precision. This shows the importance of accuracy and step-by-step problem-solving in mathematics. Great job, guys!

Conclusion

Wrapping things up, we've successfully evaluated the expression $0.677+(-0.917--0.731-0.814)^3$ , and our final answer is $-0.323$ . Remember, the key to tackling these kinds of problems is to break them down into smaller, manageable steps. We focused on simplifying inside the parentheses first, then dealt with the exponent, and finally performed the addition. Throughout the process, we paid close attention to the order of operations (PEMDAS/BODMAS) and the rules for adding and subtracting negative numbers. By understanding these fundamental principles, we can confidently approach more complex mathematical challenges. Keep practicing, and you'll become a math whiz in no time! If you guys have any questions, feel free to ask. Happy calculating!