Simplify Complex Math Expressions: Step-by-Step Solutions
Hey guys! Let's dive into simplifying some complex mathematical expressions. We've got a set of problems here that might look intimidating at first glance, but don't worry, we'll break them down step by step. Remember the order of operations (PEMDAS/BODMAS) – Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). Stick with me, and you'll be a pro at simplifying these in no time!
1. 18 - [4 - (6 - (3 + 2 of 6 - 15))]
When tackling these complex mathematical expressions, the key is to focus on the innermost parentheses first and work your way outwards. This ensures that we maintain the correct order of operations and don't get tangled up in the numbers. In this particular expression, we have nested brackets and parentheses, so we'll start with the innermost set: (3 + 2 of 6 - 15).
Let's break it down further. The term '2 of 6' means 2 multiplied by 6, which equals 12. Now our innermost expression looks like this: (3 + 12 - 15). Next, we perform addition and subtraction from left to right. 3 + 12 equals 15, and then 15 - 15 equals 0. So, the innermost parenthesis simplifies to 0. This makes our original expression much simpler:
18 - [4 - (6 - 0)]
Now, we move to the next set of parentheses: (6 - 0), which is simply 6. Our expression now becomes:
18 - [4 - 6]
Next, we deal with the brackets: [4 - 6]. Subtracting 6 from 4 gives us -2. So, the expression simplifies further to:
18 - (-2)
Finally, subtracting a negative number is the same as adding its positive counterpart. Thus, 18 - (-2) becomes 18 + 2, which equals 20. Therefore, the simplified value of the entire expression is 20. This methodical approach of working from the inside out ensures accuracy and makes even the most daunting expressions manageable. Remember, practice makes perfect, so the more you work through these types of problems, the more comfortable you'll become with them. Keep an eye on those signs and order of operations!
2. 27 - [18 - {16 - (5 - 4 - 1)}]
When dealing with these kinds of intricate arithmetic problems, the key is to methodically work through each layer of parentheses and brackets. Starting with the innermost set makes sure we follow the correct order of operations, preventing errors along the way. In this expression, we first focus on the innermost parentheses: (5 - 4 - 1).
Let's break it down step by step. First, we subtract 4 from 5, which gives us 1. So, the expression inside the parentheses becomes (1 - 1). Subtracting 1 from 1 leaves us with 0. Therefore, the innermost parentheses simplify to 0. Now, we can rewrite the original expression as:
27 - [18 - {16 - 0}]
Next, we move to the curly braces { }. Inside the curly braces, we have 16 - 0, which is simply 16. So, the expression now looks like this:
27 - [18 - 16]
Now, we focus on the brackets [ ]. Inside the brackets, we have 18 - 16, which equals 2. The expression is now simplified to:
27 - 2
Finally, we subtract 2 from 27, which gives us 25. Therefore, the simplified value of the entire expression is 25. By systematically working from the innermost parentheses outward, we ensure that we follow the correct order of operations. This approach not only makes the problem easier to manage but also reduces the chance of making mistakes. Always remember to double-check each step, especially when dealing with multiple layers of parentheses and brackets, to maintain accuracy and build confidence in your calculations.
3. 56 - 42 + 3 of {8 (20 + 4)} + 4 x 2
Okay, guys, let's tackle this one! To solve this math problem, it's super important to remember our order of operations. That's PEMDAS (or BODMAS, depending on where you learned it!), which tells us to handle Parentheses/Brackets first, then Exponents/Orders, Multiplication and Division (from left to right), and finally Addition and Subtraction (also from left to right). Got it? Great! So, looking at our expression, 56 - 42 + 3 of {8 (20 + 4)} + 4 x 2, the first thing we need to deal with is the parentheses.
Inside the parentheses, we have (20 + 4), which is a straightforward 24. Now, the expression looks like this: 56 - 42 + 3 of {8 * 24} + 4 x 2. Next up are the curly braces { }. Inside, we have 8 multiplied by 24. Doing that math gives us 192. So, our expression now reads: 56 - 42 + 3 of 192 + 4 x 2.
Now, we come to the "of" part. In this context, "3 of 192" means 3 multiplied by 192. When we do that, we get 576. The expression has transformed into: 56 - 42 + 576 + 4 x 2. Next, we handle the multiplication: 4 x 2, which equals 8. Our expression is getting simpler: 56 - 42 + 576 + 8. Now it's all addition and subtraction, and we just work from left to right.
First, 56 - 42 equals 14. Then, 14 + 576 equals 590. Finally, 590 + 8 equals 598. So, after all that careful stepping through, the simplified answer to the expression is 598. Remember, the key is to take it one step at a time, following PEMDAS (or BODMAS) to keep everything in order. You got this!
4. 24 + {87 - 3 (5 of 5)]
Let's break down this expression step by step. This mathematical simplification requires us to adhere strictly to the order of operations, commonly remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). Alternatively, you might know it as BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction). Either way, the principle remains the same: we must tackle operations in the correct sequence to arrive at the accurate answer. In our expression, 24 + {87 - 3 (5 of 5)], we see parentheses and curly braces, and we'll need to handle them systematically.
First, let's focus on what's inside the curly braces: {87 - 3 (5 of 5)]. Within this, we have both a subtraction and a term that says "3 (5 of 5)". The "of" here implies multiplication, so "5 of 5" means 5 multiplied by 5, which equals 25. Now, our expression inside the curly braces looks like this: {87 - 3 * 25]. We still need to address the multiplication before the subtraction.
Next, we multiply 3 by 25, which gives us 75. So, within the curly braces, we now have {87 - 75]. Subtracting 75 from 87, we get 12. The curly braces, therefore, simplify to 12. This simplifies our entire original expression to: 24 + 12. Finally, we simply add 24 and 12, which gives us 36. So, the simplified value of the expression 24 + {87 - 3 (5 of 5)] is 36. Remember, paying close attention to the order of operations is crucial in these types of problems. Taking it one step at a time and carefully evaluating each operation will lead you to the correct solution every time. Keep practicing, and you'll become a pro at these in no time!
5. {5 (18 + 8 - 5) - 30} + 2 × 10 + 5
Alright, let's dive into this math problem! Just like before, we need to keep our eye on the order of operations to make sure we get this right. Remember PEMDAS (or BODMAS)? It's our trusty guide! So, looking at the expression {5 (18 + 8 - 5) - 30} + 2 × 10 + 5, we see parentheses and multiplication, so we know where to start.
First up, we've got the parentheses: (18 + 8 - 5). Inside here, we do addition and subtraction from left to right. So, 18 + 8 is 26, and then 26 - 5 is 21. This means our parentheses simplify down to 21. Now, the expression looks like this: {5 * 21 - 30} + 2 × 10 + 5. We still have those curly braces to deal with, so let's stay focused.
Inside the curly braces, we have 5 multiplied by 21, which gives us 105. So, now we have {105 - 30}. Subtracting 30 from 105 gives us 75. So, the curly braces simplify to 75. Our expression is getting cleaner: 75 + 2 × 10 + 5. Next, we handle any multiplication. We have 2 multiplied by 10, which is 20. So, the expression becomes: 75 + 20 + 5.
Now, it's all addition, which is the easy part! We just add from left to right. 75 + 20 is 95, and then 95 + 5 is 100. So, after carefully working through all the steps, the simplified answer to this expression is 100. The secret to these problems is to take them slowly, one step at a time, and always follow the order of operations. Keep up the great work!
6. -5 {- (48 + 16) + 2} × 6
Okay, let's break down this numerical expression together! Remember our friend PEMDAS (or BODMAS)? It's super important here to keep everything in order. Our expression is -5 {- (48 + 16) + 2} × 6, and we see parentheses, curly braces, multiplication, and those pesky negative signs. No sweat, we've got this!
First, let's tackle those parentheses: (48 + 16). Adding 48 and 16 gives us 64. So, the expression inside the curly braces now looks like this: {- 64 + 2}. Notice that negative sign outside the parentheses? It applies to the whole result of the parentheses. Now, we add -64 and 2, which gives us -62. So, the curly braces simplify to -62.
Our expression now looks like this: -5 * -62 × 6. Next up, we just have multiplication. Remember, multiplying two negative numbers gives a positive, so let's keep that in mind. First, let's multiply -5 and -62. That gives us 310. Now, we have 310 × 6. Multiplying 310 by 6 gives us 1860. So, after all the careful steps, the simplified answer to this expression is 1860. Remember, the key is to take it one piece at a time, keeping a close eye on those signs, and following the order of operations. You're doing great!
7. 81 + 9 of (3 x 2) × 5 of 6
Alright, let's get our teeth into this equation simplification! We've got 81 + 9 of (3 x 2) × 5 of 6 to deal with. Remember PEMDAS (or BODMAS)? It's going to be our best friend here! We've got parentheses, "of" (which means multiplication), and regular multiplication and addition. Let's take it step by step and make sure we get it right.
First up, we need to deal with what's inside the parentheses: (3 x 2). That's a straightforward one – 3 multiplied by 2 is 6. So, we can replace the parentheses with 6. Now, our expression looks like this: 81 + 9 of 6 × 5 of 6. Next, we tackle the "of" operations. Remember, "of" in this context means multiplication.
So, let's do "9 of 6" first. That's 9 multiplied by 6, which gives us 54. Now, we have: 81 + 54 × 5 of 6. Let's do the next "of": "5 of 6". That's 5 multiplied by 6, which gives us 30. So, the expression now looks like this: 81 + 54 × 30. We're getting there! Now, we need to do the multiplication before the addition.
We've got 54 multiplied by 30. If we multiply those together, we get 1620. So, our expression is now super simple: 81 + 1620. Finally, we just need to add 81 and 1620. When we add those together, we get 1701. So, after carefully following the order of operations, we've simplified the expression all the way down to 1701. Great job! Remember, slow and steady wins the race with these problems. Keep up the fantastic work!
8. 22 - (4) - 5 - (-48) + (-16 + 9 - 7 + 3)
Okay, let's break down this mathematical problem! We've got a mix of positive and negative numbers, parentheses, and a bunch of additions and subtractions. But don't worry, we can handle it! Remember, the key is to take it step by step and pay close attention to those signs. Our expression is 22 - (4) - 5 - (-48) + (-16 + 9 - 7 + 3).
First things first, let's deal with those parentheses. We've got (4), which is just 4, and (-48), which is negative 48. We also have (-16 + 9 - 7 + 3), which we need to simplify. Inside this parenthesis, we'll just go from left to right. So, -16 + 9 is -7. Then, -7 - 7 is -14. Finally, -14 + 3 is -11. So, that whole parenthesis simplifies to -11.
Now, let's rewrite our expression with the simplified parentheses: 22 - 4 - 5 - (-48) + (-11). Next, let's handle that subtraction of a negative number. Remember, subtracting a negative is the same as adding a positive. So, - (-48) becomes + 48. Our expression now looks like this: 22 - 4 - 5 + 48 + (-11).
Now, let's just work from left to right, doing the additions and subtractions. 22 - 4 is 18. Then, 18 - 5 is 13. Next, 13 + 48 is 61. Finally, 61 + (-11) is the same as 61 - 11, which gives us 50. So, after carefully working through each step, the simplified answer to this expression is 50. High five! You nailed it! Remember, these problems are all about taking your time and paying attention to the details. The more you practice, the easier they'll become!
