Solving $48 \div 6 \times 2^2-(3+5)$ A Step-by-Step Guide

Hey guys! Ever stumbled upon a math problem that looks like a jumbled mess of numbers and operations? You're not alone! These types of problems often involve the order of operations, which is a set of rules that dictate the sequence in which we perform mathematical calculations. Today, we're going to break down a classic example: $48 \div 6 \times 2^2-(3+5)$. Buckle up, because we're about to embark on a mathematical adventure!

Understanding the Order of Operations (PEMDAS/BODMAS)

Before we even think about tackling our equation, it's crucial to understand the order of operations. This is our roadmap, ensuring we arrive at the correct answer. Many of us were taught an acronym to remember the order, either PEMDAS or BODMAS. Let's dissect both:

• PEMDAS:
  • Parentheses
  • Exponents
  • Multiplication and Division (from left to right)
  • Addition and Subtraction (from left to right)
• BODMAS:
  • Brackets
  • Orders (powers and square roots, etc.)
  • Division and Multiplication (from left to right)
  • Addition and Subtraction (from left to right)

Notice that both acronyms essentially convey the same message. The key takeaway is that parentheses/brackets come first, then exponents/orders, followed by multiplication and division (performed from left to right), and finally addition and subtraction (also from left to right). This left-to-right rule for multiplication/division and addition/subtraction is super important because it ensures we handle operations in the correct sequence when they appear at the same level.

So, why is this order so important? Imagine if we just performed the operations from left to right without any rules. We'd get a completely different answer! The order of operations provides a universal standard, ensuring everyone arrives at the same solution, no matter who's doing the math. It's the mathematical equivalent of a traffic light, preventing chaos and ensuring a smooth flow of calculations.

Think of it like building a house. You wouldn't start painting the walls before laying the foundation, right? Similarly, in math, certain operations need to be performed before others to ensure a logical and accurate result. Ignoring the order of operations is like trying to put the roof on before the walls are up – it's just not going to work!

Step-by-Step Solution of $48 \div 6 \times 2^2-(3+5)$

Now that we've solidified our understanding of the order of operations, let's apply it to our problem: $48 \div 6 \times 2^2-(3+5)$. We'll break it down step-by-step, just like a math detective solving a mystery.

1. Parentheses First

According to PEMDAS/BODMAS, we need to tackle anything inside parentheses first. In our equation, we have $(3+5)$. This is a straightforward addition problem:

$3 + 5 = 8$

So, we can replace $(3+5)$ with $8$, giving us a simplified equation:

$48 \div 6 \times 2^2 - 8$

2. Exponents Next

Now that the parentheses are dealt with, we move on to exponents. We have $2^2$, which means 2 raised to the power of 2 (or 2 squared). This is simply $2 \times 2$:

$2^2 = 4$

Substituting this back into our equation, we get:

$48 \div 6 \times 4 - 8$

3. Multiplication and Division (Left to Right)

This is where the left-to-right rule comes into play. We have both division and multiplication, so we perform them in the order they appear from left to right. First, we have $48 \div 6$:

$48 \div 6 = 8$

Now our equation looks like this:

$8 \times 4 - 8$

Next, we perform the multiplication: $8 \times 4$:

$8 \times 4 = 32$

Our equation is now even simpler:

$32 - 8$

4. Addition and Subtraction (Left to Right)

Finally, we're left with subtraction. We simply subtract 8 from 32:

$32 - 8 = 24$

And there you have it! The solution to $48 \div 6 \times 2^2-(3+5)$ is 24.

Common Mistakes to Avoid

The order of operations might seem straightforward, but it's easy to slip up if you're not careful. Here are some common mistakes to watch out for:

• Forgetting the Left-to-Right Rule: This is a big one! When dealing with multiplication and division (or addition and subtraction), always work from left to right. Don't just perform the operation that looks easier first.
• Ignoring Parentheses: Parentheses are like VIPs in the math world. They demand your attention first. Don't skip them!
• Misunderstanding Exponents: Remember that an exponent tells you how many times to multiply the base by itself. $2^2$ is $2 \times 2$, not $2 \times 2$.
• Rushing Through the Steps: Take your time! Break the problem down into smaller, manageable steps. It's better to be accurate than fast.

To avoid these mistakes, it's helpful to write out each step clearly, as we did above. This allows you to track your progress and catch any errors along the way. Practice makes perfect, so the more you work with the order of operations, the more confident you'll become.

Real-World Applications of the Order of Operations

You might be thinking,