Step By Step Guide Calculating Mathematical Expressions

Hey guys! Let's dive into calculating some expressions. We'll break down each step to make sure you've got it down. No stress, just clear math!

a. $\frac{3(2+6)}{2}$

First things first, let's tackle our first expression: $\frac{3(2+6)}{2}$. The key here is following the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This order ensures we solve the expression correctly, step by step, preventing any mix-ups along the way.

Step 1 Parentheses

Within the parentheses, we have $2 + 6$. So, let's add those up first. It's like grouping things together before we deal with the outside stuff. So, $2 + 6$ becomes 8. Now our expression looks a little simpler: $\frac{3(8)}{2}$. This is a solid start, and we are on the right track to simplifying the expression further. Simplifying inside the parentheses is always the initial move, setting the stage for the next operations.

Step 2 Multiplication

Next up, we've got $3(8)$ in the numerator. Remember, the parentheses next to the 3 mean multiplication. So, we're multiplying 3 by 8. Doing the math, $3 \times 8$ equals 24. Our expression now looks even cleaner: $\frac{24}{2}$. It's getting there, right? We've handled the parentheses and the multiplication, paving the way for the final operation.

Step 3 Division

Okay, we're in the home stretch! We've got $\frac{24}{2}$, which means 24 divided by 2. This is a straightforward division problem. When we divide 24 by 2, we get 12. And that's our final answer! The value of the expression $\frac{3(2+6)}{2}$ is 12. See? Not too shabby when we break it down step by step.

In summary, we began with the original expression, simplified the addition within the parentheses, proceeded to multiplication, and concluded with division. This methodical approach not only provides the correct answer but also ensures a clear understanding of each step involved. Math isn't so scary when you take it piece by piece, right?

b. $\frac{1}{2}(14)(5)$

Now, let's move onto our next challenge: $\frac{1}{2}(14)(5)$. This one involves fractions and multiplication, but don't sweat it, we've got this! It's all about taking it one step at a time, just like we did before. Remember, order of operations is our trusty guide here. We'll glide through this just fine.

Step 1 Multiplication (Left to Right)

Since we only have multiplication here, we can just go from left to right. Easy peasy! First up, we're looking at $\frac{1}{2}(14)$. You can think of this as half of 14, or 14 multiplied by $\frac{1}{2}$. What's half of 14? It's 7! So, $\frac{1}{2}(14)$ simplifies to 7. Now our expression is looking even simpler: $7(5)$. Isn't it satisfying how things simplify step by step?

Step 2 Multiplication

Alright, we're at the final step for this one. We've got $7(5)$, which means 7 multiplied by 5. This is a classic multiplication problem. If you know your times tables, you'll know that $7 \times 5$ is 35. And there we have it! The value of the expression $\frac{1}{2}(14)(5)$ is 35. You nailed it!

So, just to recap, we kicked things off by handling the leftmost multiplication, which led us to finding half of 14. We then wrapped things up by multiplying our intermediate result by 5. This problem really highlights how following the order of operations helps us break down expressions into manageable chunks. Who knew math could be so smooth?

c. $7^2-5^2$

Let's jump into our third expression: $7^2-5^2$. This one has exponents, which might sound intimidating, but trust me, we'll handle them like pros! Remember, the name of the game is still PEMDAS – keeping that order of operations in mind helps us tackle any expression with confidence.

Step 1 Exponents

Alright, the first thing we need to deal with are those exponents. Remember, an exponent tells us how many times to multiply the base by itself. So, $7^2$ means 7 multiplied by 7, and $5^2$ means 5 multiplied by 5. Let's calculate these separately.

• $7^2$ is $7 \times 7$, which equals 49. Great! Now we know what $7^2$ is.
• Next, we calculate $5^2$, which is $5 \times 5$, giving us 25. Fantastic! We've cracked both exponents.

Now we can rewrite our expression, replacing the exponents with their values: $49 - 25$. See how much simpler it looks already? Taking care of those exponents really cleans things up.

Step 2 Subtraction

Okay, we're almost there! Now we just have one simple subtraction problem to solve: $49 - 25$. This is just straightforward subtraction. When we subtract 25 from 49, we get 24. Boom! We've got our answer. The value of the expression $7^2-5^2$ is 24. You're doing awesome!

In short, we started with the expression, tackled the exponents one by one, and then finished up with a straightforward subtraction. This example is a perfect showcase of how exponents fit into the order of operations and how understanding them can make expressions much less daunting. Keep up the great work!

d. $17-6 \cdot 2+4 \div 2$

Last but not least, we have $17-6 \cdot 2+4 \div 2$. This expression mixes subtraction, multiplication, addition, and division – a real mathematical potpourri! But don't worry, we've got our trusty PEMDAS to guide us through. It's all about knowing which operation to handle first, and we've got this down.

Step 1 Multiplication and Division (Left to Right)

According to PEMDAS, we need to tackle multiplication and division before we even think about addition or subtraction. But here's a little twist: when we have both multiplication and division, we do them in the order they appear from left to right. It's like reading a sentence; we go from left to right to make sense of it.

• First up, we have $6 \cdot 2$, which is 6 multiplied by 2. That's 12. So, we replace $6 \cdot 2$ with 12 in our expression. Now it looks like this: $17 - 12 + 4 \div 2$.
• Next, we've got the division: $4 \div 2$. This is 4 divided by 2, which equals 2. So, we replace $4 \div 2$ with 2 in our expression. Now we're looking at: $17 - 12 + 2$. Getting cleaner and cleaner, right?

Step 2 Addition and Subtraction (Left to Right)

Now we're down to just addition and subtraction. Similar to multiplication and division, we handle these in the order they appear from left to right. So, we'll start with the leftmost operation, which is subtraction.

• We have $17 - 12$. Subtracting 12 from 17 gives us 5. So, now our expression is simply: $5 + 2$.
• Finally, we have $5 + 2$, which is 5 plus 2. This equals 7. And that's it! The value of the expression $17-6 \cdot 2+4 \div 2$ is 7. High five!

To sum up, we meticulously followed the order of operations, starting with multiplication and division from left to right, and then moving onto addition and subtraction, also from left to right. This example really highlights why sticking to PEMDAS is crucial for getting the correct answer in more complex expressions. You handled it like a champ!

Conclusion

So, there you have it! We've walked through calculating the values of four different expressions, breaking down each step along the way. Whether it was dealing with parentheses, exponents, or a mix of operations, the key was to take it slow, follow the order of operations, and tackle one step at a time. You've shown that even complex expressions can be manageable and, dare I say, even fun when you approach them with a clear strategy. Keep practicing, and you'll be a math whiz in no time! You got this!