Math Expression Evaluation: A Step-by-Step Guide
Hey math whizzes and number crunchers! Today, we're diving deep into a super interesting math problem that'll test your skills with absolute values, multiplication, and basic arithmetic. We're going to evaluate the following expression: |0.741| - 0.624 × |0.97| + 0.2 = □. Don't worry if it looks a bit daunting at first, guys. We'll break it down step-by-step, making sure you understand every single part of the process. By the end of this, you'll be a pro at simplifying expressions like this!
Understanding Absolute Value: The First Hurdle
So, what's the deal with these vertical lines, | |, in our expression? That, my friends, is the absolute value. Think of it as a number's distance from zero on the number line. No matter if the number is positive or negative, its absolute value is always positive. For example, the absolute value of 5, written as |5|, is just 5. And the absolute value of -5, written as |-5|, is also 5. It's like a number's ego – it always wants to be seen in its best, positive light!
In our specific problem, we have two instances of absolute value: |0.741| and |0.97|. Since both 0.741 and 0.97 are already positive numbers, their absolute values are simply themselves. So, |0.741| is just 0.741, and |0.97| is 0.97. This is a crucial first step, as it simplifies the expression significantly. We can now rewrite our original expression, replacing the absolute value parts with their positive counterparts:
0.741 - 0.624 × 0.97 + 0.2 = □
See? Already looks a lot less intimidating, right? This is the power of understanding the fundamental concepts. Always tackle the absolute values first, and you're well on your way to conquering the rest of the problem. Remember, in mathematics, just like in life, breaking down complex challenges into smaller, manageable steps makes all the difference. Keep this principle in mind as we move forward, because we're not done yet!
Order of Operations: PEMDAS to the Rescue!
Now that we've handled the absolute values, we need to think about the order in which we perform the remaining operations: subtraction, multiplication, and addition. This is where the trusty PEMDAS rule comes into play. PEMDAS is an acronym that stands for:
- Parentheses (or Brackets)
- Exponents
- Multiplication and Division (from left to right)
- Addition and Subtraction (from left to right)
It's the golden rule of arithmetic that ensures everyone gets the same answer when evaluating an expression. If we don't follow PEMDAS, chaos would ensue, and math would be a very different, very confusing subject!
Looking at our simplified expression, 0.741 - 0.624 × 0.97 + 0.2, we can see we have subtraction, multiplication, and addition. According to PEMDAS, multiplication comes before addition and subtraction. Therefore, the very next step is to calculate the product of 0.624 and 0.97.
Let's do that multiplication:
0.624 × 0.97
To make this easier, you can line up the decimals and multiply as if they were whole numbers, then place the decimal point in the correct spot in the answer.
0.624
× 0.97
-------
4368 (0.624 × 7)
56160 (0.624 × 90)
-------
60528
Now, count the total number of decimal places in the numbers being multiplied. 0.624 has three decimal places, and 0.97 has two decimal places, for a total of five decimal places. So, we need to place the decimal point five places from the right in our result: 0.60528.
This single calculation, 0.624 × 0.97 = 0.60528, has significantly simplified our expression further. We can now substitute this result back into our equation:
0.741 - 0.60528 + 0.2 = â–¡
We're getting closer, folks! Each step taken according to PEMDAS brings us nearer to the final answer. It's like a treasure hunt, and each solved piece of the puzzle reveals more of the map. Keep that momentum going, and don't be afraid to grab a calculator if you need to double-check your multiplication – that's what they're there for!
Final Calculations: Addition and Subtraction
We've successfully navigated the absolute values and the multiplication step, thanks to PEMDAS. Now, our expression looks like this: 0.741 - 0.60528 + 0.2 = â–¡. The remaining operations are subtraction and addition. Remember, according to PEMDAS, addition and subtraction are performed from left to right.
So, the first operation we need to tackle is the subtraction: 0.741 - 0.60528. Let's line this up carefully, making sure the decimal points are aligned:
0.74100
- 0.60528
---------
0.13572
When subtracting, it's often helpful to add trailing zeros to the number with fewer decimal places to ensure all digits align correctly. So, 0.741 becomes 0.74100. Performing the subtraction gives us 0.13572.
Now, we substitute this result back into our expression:
0.13572 + 0.2 = â–¡
We're on the home stretch, guys! The final step is a simple addition. We need to add 0.2 to 0.13572. Again, lining up the decimals is key:
0.13572
+ 0.20000
---------
0.33572
Adding 0.2 (which is 0.20000 when we add trailing zeros for alignment) to 0.13572 gives us 0.33572.
And there you have it! The final answer to our expression |0.741| - 0.624 × |0.97| + 0.2 is 0.33572.
Simplifying the Answer: Integers or Decimals?
The problem states to "Simplify your answer. Type an integer or a decimal." Our result, 0.33572, is already in its simplest decimal form. It's not an integer because it has a fractional part. So, we can confidently say that our simplified answer is 0.33572.
Evaluating expressions like this might seem like a puzzle, but by understanding the core mathematical principles – like absolute value and the order of operations (PEMDAS) – you can solve them with confidence. It's all about taking it one step at a time and not getting overwhelmed by the whole picture. Keep practicing these types of problems, and you'll become a math ninja in no time! You guys totally got this!