Simplify: 85.6 + (28.96 - 4.1²) / 2.5

Nov 4, 2025 by Admin 38 views

Let's break down how to simplify the expression $85.6 + (28.96 - 4.1^2) div 2.5$ using the order of operations. You might remember this as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). Following these rules ensures we get to the correct answer step by step. Grab your calculators, guys, because we're about to dive in!

Step-by-Step Solution

Parentheses First: We start with what’s inside the parentheses: $(28.96 - 4.1^2)$ .
Evaluate the Exponent: Inside the parentheses, we first deal with the exponent: $4.1^2 = 4.1 cdot 4.1 = 16.81$ . So now our expression inside the parentheses looks like this: $(28.96 - 16.81)$ .
Perform the Subtraction: Now, subtract: $28.96 - 16.81 = 12.15$ . Great! The parentheses are now simplified to a single number.
Rewrite the Expression: Our original expression now becomes: $85.6 + 12.15 div 2.5$ .
Perform the Division: Next up is division: $12.15 div 2.5 = 4.86$ .
Perform the Addition: Finally, we do the addition: $85.6 + 4.86 = 90.46$ .

So, the simplified value of the expression is $90.46$ .

Detailed Breakdown

Initial Expression

We start with the expression:

$85.6 + (28.96 - 4.1^2) div 2.5$

Exponent Evaluation

First, let's evaluate the exponent:

$4.1^2 = 4.1 cdot 4.1 = 16.81$

So, the expression becomes:

$85.6 + (28.96 - 16.81) div 2.5$

Parenthetical Subtraction

Next, perform the subtraction inside the parentheses:

$28.96 - 16.81 = 12.15$

Now, substitute this back into the expression:

$85.6 + 12.15 div 2.5$

Division Operation

Perform the division:

$12.15 div 2.5 = 4.86$

Substitute this result back:

$85.6 + 4.86$

Final Addition

Finally, perform the addition:

$85.6 + 4.86 = 90.46$

Therefore, the simplified expression is $90.46$ .

Why Order of Operations Matters

The order of operations is super important in math because it ensures everyone solves the same problem in the same way, getting the same answer. Without it, math would be chaos! Imagine if one person decided to add before dividing – they’d get a completely different result. PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) is the golden rule. First, we tackle anything inside parentheses. Then, we deal with exponents. After that, it’s multiplication and division (from left to right), and finally, addition and subtraction (again, from left to right). By sticking to this order, we maintain consistency and accuracy in our calculations.

For instance, let’s say we ignored the order of operations and added $85.6 + 12.15$ first in our original problem. We'd get $97.75$ , and then dividing that by $2.5$ would give us a completely wrong answer. So, always remember PEMDAS to keep your math on the right track!

Practical Applications

You might be wondering, “Where would I ever use this in real life?” Well, understanding the order of operations is actually quite useful! Think about calculating your taxes, figuring out discounts while shopping, or even when you're coding. Many formulas in finance, science, and engineering rely on the correct order of operations to produce accurate results. For example, when calculating compound interest, you need to follow the order of operations to determine the final amount accurately.

Consider a simple example: You want to buy three items that cost $10 each, and you have a coupon for $5 off your total purchase. The expression would be $(3 cdot 10) - 5$ . If you subtracted before multiplying, you'd get $3 cdot 5 = 15$ , which is incorrect. The correct calculation is $30 - 5 = 25$ . This shows how vital it is to follow PEMDAS in everyday scenarios to avoid mistakes.

Common Mistakes to Avoid

One common mistake is forgetting to handle exponents before multiplication or division. Another is mixing up the order of addition and subtraction or multiplication and division—remember to work from left to right for those! Also, be super careful with signs (positive and negative numbers), especially when dealing with parentheses. It's easy to make a small error that throws off the entire calculation.

To avoid these pitfalls, always double-check your work and take it one step at a time. Writing out each step clearly can help you catch errors before they snowball. And don’t hesitate to use a calculator to verify your answers, especially for complex expressions. Practice makes perfect, so the more you work with these types of problems, the more comfortable and accurate you’ll become!

Rounding to the Nearest Hundredth

In this particular problem, our final answer was already to the nearest hundredth ( $90.46$ ), so we didn't need to do any additional rounding. However, it’s always good to understand how rounding works. When rounding to the nearest hundredth, you look at the digit in the thousandths place. If it's 5 or greater, you round up the hundredths digit. If it's less than 5, you leave the hundredths digit as it is.

For example, if our answer had been $90.465$ , we would round it up to $90.47$ . If it were $90.464$ , we would leave it as $90.46$ . Rounding ensures that our answer is precise to the required level of accuracy, which is particularly important in fields like finance and engineering where even small differences can have significant impacts.

Practice Problems

To solidify your understanding, try simplifying these expressions using the order of operations. Remember PEMDAS, and take your time!

$15.2 + (3.5^2 - 8.1) div 2$
$100 - (5.2 cdot 4.8 + 12.4) div 3.1$
$42.6 div (2.1 + 3.9) - 1.5^2$

Work through each problem step by step, and then check your answers. The more you practice, the easier it will become to apply the order of operations correctly and efficiently. And remember, it’s okay to make mistakes—that’s how we learn!

By following these steps and keeping the order of operations in mind, you can confidently simplify complex expressions. Stick with it, and you’ll become a math whiz in no time!