Decoding Mathematical Expressions A Step-by-Step Guide To Solving (7 X 6) ÷ 2[{45 ÷ 3(7 X 2-15+6)}]+4

Hey math enthusiasts! Today, we're going to dissect a seemingly complex mathematical expression. Don't worry, we'll break it down step-by-step, making sure everyone understands the underlying principles. We're tackling this beast: $(7 \times 6) \div 2[\{45 \div 3(7 \times 2-15+6)\}]+4$. Buckle up, it's going to be a fun ride!

Understanding the Order of Operations (PEMDAS/BODMAS)

Before we even think about plugging in numbers, let's quickly revisit the order of operations. This is the golden rule that dictates how we solve mathematical expressions. You might have heard of the acronyms PEMDAS or BODMAS. They stand for:

• Parentheses / Brackets
• Exponents / Orders
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

This order ensures we all arrive at the same answer, no matter how complex the expression looks. Ignoring this order will lead to mathematical chaos, and nobody wants that! So, keep PEMDAS/BODMAS in the back of your mind as we proceed.

Step-by-Step Breakdown of the Expression

Now, let's apply the order of operations to our expression: $(7 \times 6) \div 2[\{45 \div 3(7 \times 2-15+6)\}]+4$. We'll take it one step at a time.

1. Innermost Parentheses First

Our first task is to conquer the innermost parentheses: $(7 \times 2-15+6)$.

• Multiplication: We start with the multiplication within the parentheses: $7 \times 2 = 14$. So, the expression becomes $(14 - 15 + 6)$.
• Subtraction: Next up is subtraction: $14 - 15 = -1$. Now we have $(-1 + 6)$.
• Addition: Finally, we add: $-1 + 6 = 5$.

So, the innermost parentheses simplify to 5. Let's rewrite the entire expression with this simplification: $(7 \times 6) \div 2[\{45 \div 3(5)\}]+4$.

2. Curly Braces

Now we move to the curly braces: $\{45 \div 3(5)\}$. Remember, multiplication and division have equal priority, so we perform them from left to right.

• Division: First, we divide: $45 \div 3 = 15$. The expression within the curly braces now looks like this: $15(5)$.
• Multiplication: Next, we multiply: $15 \times 5 = 75$.

The curly braces simplify to 75. Our expression now looks like this: $(7 \times 6) \div 2[75]+4$.

3. Square Brackets

Next in line are the square brackets: $2[75]$. This simply means $2 \times 75$.

• Multiplication: $2 \times 75 = 150$.

Our expression is becoming much simpler: $(7 \times 6) \div 150 + 4$.

4. Parentheses

Now we tackle the remaining parentheses: $(7 \times 6)$.

• Multiplication: $7 \times 6 = 42$.

We're getting there! The expression is now: $42 \div 150 + 4$.

5. Division

Time for division: $42 \div 150$. This can be expressed as a fraction, $\frac{42}{150}$, which simplifies to $\frac{7}{25}$ or as a decimal, 0.28.

Our expression is looking lean and mean: $0.28 + 4$.

6. Addition

Finally, we add: $0.28 + 4 = 4.28$.

The Grand Finale: The Solution

Phew! We've navigated the twists and turns of the order of operations and arrived at our final answer:

$(7 \times 6) \div 2[{45 \div 3(7 \times 2-15+6)}]+4 = 4.28

Congratulations, you've successfully solved a complex mathematical expression! Remember, the key is to take it one step at a time, following the order of operations.

Common Pitfalls and How to Avoid Them

Guys, even with PEMDAS/BODMAS as our guide, it's easy to stumble. Let's look at some common mistakes people make and how to avoid them.

• Forgetting the Order: This is the biggest culprit! Many errors happen when we don't strictly adhere to PEMDAS/BODMAS. Always double-check the order before you start crunching numbers.
• Incorrectly Handling Multiplication and Division (or Addition and Subtraction): Remember, these operations have equal priority. We work from left to right. Don't jump ahead and multiply before dividing (or subtract before adding) unless parentheses dictate it.
• Sign Errors: Pay close attention to negative signs! A misplaced negative can throw off the entire calculation. Be especially careful when dealing with subtraction and distribution.
• Rushing Through the Steps: Math isn't a race. Take your time, write down each step clearly, and double-check your work. It's better to be accurate than fast.
• Ignoring Parentheses: Parentheses are your best friends! They tell you exactly what to do first. Don't skip over them or misinterpret their groupings.

To avoid these pitfalls, practice makes perfect! The more you work through problems, the more comfortable you'll become with the order of operations.

Practice Problems to Sharpen Your Skills

Okay, now that we've conquered that beast of an expression, let's flex our mathematical muscles with a few practice problems. Remember, the key is to apply PEMDAS/BODMAS diligently. Work through these on your own, and then we can compare notes.

Here are a few to get you started:

1. $10 + 2 \times (15 - 3) \div 4$
2. $(8 + 4) \div 2 - 3 \times 2 + 1$
3. $3^2 + 16 \div (5 - 3) \times 2$
4. $[12 - (3 + 1) \times 2] + 5 \times 3$
5. $48 \div [6 \times (8 - 4)] + 2^3 - 1$

Work through these carefully, showing each step. This will help you solidify your understanding of the order of operations. Don't be afraid to make mistakes – that's how we learn! The important thing is to understand why you made a mistake and how to correct it.

Real-World Applications of Order of Operations

You might be thinking,