Step-by-Step Guide Calculating Expression Values

Hey guys! Let's dive into some math problems and break down how to calculate the values of different expressions. We'll go through each step, so you can easily follow along. Get ready to sharpen your pencils and your minds!

Expression A: $2 \cdot(3(5+2)-1)$

Our first expression is a complex-looking one, but don't worry, we'll tackle it piece by piece. Remember the golden rule in math: the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). Let's carefully apply this principle here.

Step 1: Parentheses First

The innermost parentheses are our first target. We have (5+2) inside the larger parentheses. Let's add those together:

5 + 2 = 7

So, now our expression looks like this:

2 \cdot (3(7) - 1)

Step 2: Multiplication Within Parentheses

Next up, we deal with the multiplication inside the parentheses. We have 3(7), which means 3 multiplied by 7:

3 * 7 = 21

Our expression now becomes:

2 \cdot (21 - 1)

Step 3: Subtraction Within Parentheses

We still have some work to do inside the parentheses. Let's subtract 1 from 21:

21 - 1 = 20

Now, the expression simplifies to:

2 \cdot 20

Step 4: Final Multiplication

Finally, we perform the last multiplication:

2 * 20 = 40

So, the value of the expression $2 \cdot(3(5+2)-1)$ is 40. See? Not so scary when we break it down!

Expression B: $6-2(4+5)+6$

Moving on to our second expression, we'll again use the order of operations to guide us. This expression involves subtraction, multiplication, addition, and parentheses, so let's get to it! Understanding and correctly implementing the order of operations is the key to solving mathematical expressions accurately.

Step 1: Parentheses

First up, the parentheses! We have (4+5) inside the parentheses. Let's add those numbers:

4 + 5 = 9

Our expression now looks like this:

6 - 2(9) + 6

Step 2: Multiplication

Next, we handle the multiplication. We have 2(9), which means 2 multiplied by 9:

2 * 9 = 18

So, our expression becomes:

6 - 18 + 6

Step 3: Subtraction and Addition (from left to right)

Now we have subtraction and addition. Remember, when faced with both, we work from left to right. First, let's subtract 18 from 6:

6 - 18 = -12

Our expression now reads:

-12 + 6

Finally, let's add 6 to -12:

-12 + 6 = -6

Thus, the value of the expression $6-2(4+5)+6$ is -6. Awesome job!

Expression C: $3 \cdot 8 \div 2^2+1$

Let's tackle the third expression, which introduces exponents into the mix! Don't worry; we'll still follow PEMDAS, giving exponents their due respect. Mastering the order of operations is crucial for accurate mathematical calculations.

Step 1: Exponents

First things first, we need to deal with the exponent. We have $2^2$, which means 2 raised to the power of 2 (2 squared):

2^2 = 2 * 2 = 4

Our expression now looks like this:

3 \cdot 8 \div 4 + 1

Step 2: Multiplication and Division (from left to right)

Next, we handle multiplication and division. Remember, we work from left to right. First, let's multiply 3 by 8:

3 * 8 = 24

Our expression becomes:

24 \div 4 + 1

Now, let's divide 24 by 4:

24 \div 4 = 6

Our expression simplifies to:

6 + 1

Step 3: Addition

Finally, we perform the addition:

6 + 1 = 7

So, the value of the expression $3 \cdot 8 \div 2^2+1$ is 7. You're getting the hang of this!

Expression D: $5-2 \cdot 3+6(3^2+1)$

Alright, let's move on to our final expression. This one packs a punch with exponents, multiplication, addition, subtraction, and parentheses! But by now, we are confident in the order of operations and how it guides us through these problems. Successfully navigating complex expressions hinges on a solid understanding of mathematical principles.

Step 1: Parentheses

As always, we start with the parentheses. Inside the parentheses, we have an exponent and addition. Let's tackle the exponent first.

Sub-step 1: Exponent within Parentheses

We have $3^2$, which means 3 raised to the power of 2 (3 squared):

3^2 = 3 * 3 = 9

Now, the expression within the parentheses becomes:

9 + 1

Sub-step 2: Addition within Parentheses

Let's add 9 and 1:

9 + 1 = 10

Our main expression now looks like this:

5 - 2 \cdot 3 + 6(10)

Step 2: Multiplication

Next up, we handle all the multiplications. We have two multiplications to take care of: 2 * 3 and 6(10).

Sub-step 1: First Multiplication

2 * 3 = 6

Sub-step 2: Second Multiplication

6 * 10 = 60

Our expression now looks like this:

5 - 6 + 60

Step 3: Subtraction and Addition (from left to right)

Finally, we tackle the subtraction and addition, working from left to right. First, let's subtract 6 from 5:

5 - 6 = -1

Our expression becomes:

-1 + 60

Now, let's add 60 to -1:

-1 + 60 = 59

Therefore, the value of the expression $5-2 \cdot 3+6(3^2+1)$ is 59. You nailed it!

Conclusion

Great job, everyone! We've successfully calculated the values of four different expressions by carefully following the order of operations. Remember, the key is to break down complex problems into smaller, manageable steps. Keep practicing, and you'll become math masters in no time! If you enjoyed this breakdown, make sure to try out some more expressions on your own. You've got this!