Mastering Mathematical Expressions A Comprehensive Guide

Hey guys! Today, we're diving deep into the fascinating world of mathematical expressions. Think of this as your ultimate guide to cracking those number puzzles. We'll break down complex calculations step by step, making sure you not only understand the how but also the why behind each operation. So, grab your calculators (or your brainpower – both work!), and let's get started!

1.63: Unraveling Exponential and Arithmetic Expressions

a) 3 × 10³ + 2 × 10² + 5 × 10

Let's tackle this expression, which combines the power of exponents with simple arithmetic. The key here is understanding the order of operations (PEMDAS/BODMAS, remember?). We'll start with the exponents, then move on to multiplication, and finally, addition. This structured approach is your best friend when dealing with any mathematical expression, ensuring you get to the correct answer efficiently and without missing a step.

First, we evaluate the exponential terms. 10³ means 10 raised to the power of 3, which is 10 multiplied by itself three times (10 × 10 × 10), resulting in 1000. Similarly, 10² is 10 squared, or 10 × 10, which equals 100. Now, let's substitute these values back into our expression: 3 × 1000 + 2 × 100 + 5 × 10. See how breaking down the problem into smaller parts makes it way less intimidating?

Next up, multiplication! We've got 3 multiplied by 1000, which gives us 3000. Then, 2 multiplied by 100 equals 200, and 5 multiplied by 10 gives us 50. Our expression now looks like this: 3000 + 200 + 50. We are in the final stretch now, let’s keep pushing to the correct answer!

Finally, the easiest part: addition. We add 3000, 200, and 50 together. 3000 plus 200 is 3200, and adding 50 to that gives us a grand total of 3250. And there you have it! The solution to 3 × 10³ + 2 × 10² + 5 × 10 is 3250. Wasn’t so bad when we broke it down piece by piece, right? This method is crucial for tackling more complex equations, where keeping track of each step is key to avoiding mistakes. Always remember, math is like building with Lego bricks – each step is a piece of the puzzle, and the order matters!

b) 35 - 2 × 1¹¹¹ + 3 × 7 × 7²

Okay, let's jump into another exciting expression! This one throws a few more curveballs our way with a mix of subtraction, multiplication, and exponents. But don't worry, we're going to handle it like pros. Just like before, our trusty order of operations (PEMDAS/BODMAS) will be our guiding star. This time, we've got a large exponent to deal with, but don't let it intimidate you – we've got this!

First things first, let's simplify those exponents. We've got 1 raised to the power of 111 (1¹¹¹). Now, here's a cool math trick: 1 raised to any power is always 1. Yep, you heard it right! So, 1¹¹¹ is simply 1. And we also have 7², which means 7 multiplied by itself, giving us 49. Let’s replace these values in our expression: 35 - 2 × 1 + 3 × 7 × 49. See how simplifying the exponents made things a lot clearer?

Next in line is the multiplication. We've got 2 multiplied by 1, which is 2. Then, we have 3 multiplied by 7, which gives us 21. And finally, 21 multiplied by 49… Okay, that's a bigger number, but no sweat! It equals 1029. Let's plug those values back in: 35 - 2 + 1029. We’ve handled the exponents and the multiplications, so we are on track to simplifying the overall expression now.

Now, we're down to subtraction and addition. Remember, we perform these operations from left to right. So, let's start with 35 minus 2, which equals 33. Now, we add 1029 to 33, and that gives us a grand total of 1062. So, the final answer to 35 - 2 × 1¹¹¹ + 3 × 7 × 7² is 1062. Woohoo! You've conquered another mathematical mountain. The key takeaway here is to tackle expressions one step at a time, and those big, scary problems suddenly become a series of small, manageable tasks. Remember, patience and a methodical approach are your superpowers in the math world!

c) 5 × 4³ + 2 × 3 - 81 × 2 + 7

Alright, guys, let’s dive into another fascinating equation! This time, we have a mix of multiplication, exponents, subtraction, and addition – a true mathematical medley! But don’t worry, we're going to break it down just like we always do, using our trusty order of operations (PEMDAS/BODMAS). Remember, this is like our roadmap to solving the puzzle, ensuring we take the right turns at every step.

First up, let’s tackle those exponents. We’ve got 4³, which means 4 raised to the power of 3. That's 4 multiplied by itself three times (4 × 4 × 4), which equals 64. Now, let's substitute that value back into our expression: 5 × 64 + 2 × 3 - 81 × 2 + 7. See how much simpler it looks already? Dealing with exponents first helps to clear the path for the rest of the calculations.

Next in line is multiplication. We have a few multiplications to handle here. First, 5 multiplied by 64. If you’re quick with your mental math, you’ll know that’s 320. If not, no worries – a little side calculation does the trick! Then, we have 2 multiplied by 3, which gives us 6. And finally, 81 multiplied by 2, which equals 162. Let’s plug these results back into our equation: 320 + 6 - 162 + 7. We’ve cleared the multiplication hurdle, which means we are getting closer to the finish line.

Now, we’re left with addition and subtraction. Remember, we tackle these operations from left to right. So, let's start with 320 plus 6, which gives us 326. Then, we subtract 162 from 326. If you do the math, that leaves us with 164. Finally, we add 7 to 164, which brings us to our final answer: 171. So, 5 × 4³ + 2 × 3 - 81 × 2 + 7 equals 171. Great job! You've navigated through a pretty complex expression with confidence and skill. Remember, the key is to take it one step at a time, focusing on each operation in the correct order. Keep practicing, and you'll be a math whiz in no time!

1.64: Mastering Nested Parentheses and Exponents

a) [(33 - 3) ÷ 3]^{3+3}

Alright, let's dive into a mathematical expression that looks a bit like a puzzle within a puzzle! We've got parentheses nested inside brackets, exponents, and division all hanging out together. But don't let it intimidate you; we're going to tackle it step-by-step, just like true math detectives. Remember our trusty order of operations (PEMDAS/BODMAS)? It's going to be our best friend here, guiding us through the maze of numbers and symbols.

First things first, let’s conquer the innermost parentheses. We've got (33 - 3) sitting snug inside. That's a straightforward subtraction: 33 minus 3 equals 30. So, let’s replace that part of the expression: [30 ÷ 3]^{3+3}. See how we're simplifying things bit by bit? It’s like peeling an onion – layer by layer, we get to the heart of the problem.

Next up, we deal with what's inside the brackets. We have 30 ÷ 3. That's a simple division, and it equals 10. So, our expression now looks like this: 10^{3+3}. We're making great progress! We’ve handled the parentheses and the division within the brackets, so we’re on the right track.

Now, let's focus on the exponent. But wait, there’s a little addition hiding up there! We've got 3 + 3 as the exponent, which equals 6. So, our expression transforms into 10⁶. Ah, now it looks much more manageable! We've simplified the exponent, and now we’re ready for the final calculation.

Finally, we calculate 10 raised to the power of 6. That’s 10 multiplied by itself six times, which is 10 × 10 × 10 × 10 × 10 × 10. If you do the math, that’s a whopping 1,000,000! So, [(33 - 3) ÷ 3]^{3+3} equals 1,000,000. Woohoo! You've cracked a complex expression with nested operations. The key here is to always start from the innermost parentheses and work your way outwards, step by step. It's like untangling a knot – patience and a systematic approach will always lead you to the solution!

b) 2⁵ + 2 × {12 + 2 × [3 × (5 - 2) + 1] + 1} + 1

Okay, guys, brace yourselves! We’re about to tackle a mathematical beast – an expression loaded with parentheses, brackets, braces, multiplication, exponents, and addition. It looks like a maze, right? But don’t worry, we’re going to conquer it with our step-by-step strategy and our trusty order of operations (PEMDAS/BODMAS). Think of it as an adventure, where each step brings us closer to the treasure – the correct answer!

First things first, let's dive into the innermost layer of parentheses. We have (5 - 2) nestled deep inside. That's a simple subtraction, and it equals 3. Let’s replace that in our expression: 2⁵ + 2 × {12 + 2 × [3 × 3 + 1] + 1} + 1. See how we’ve already made progress? Tackling the innermost operations first helps to simplify the bigger picture.

Now, let's move to the brackets. Inside the brackets, we have 3 × 3 + 1. Following the order of operations, we do the multiplication first: 3 × 3 equals 9. So, the expression inside the brackets becomes 9 + 1, which equals 10. Let’s substitute that back in: 2⁵ + 2 × {12 + 2 × 10 + 1} + 1. We’ve cleared the brackets, and things are looking brighter!

Next up are the braces. Inside the braces, we have 12 + 2 × 10 + 1. Again, we follow the order of operations and do the multiplication first: 2 × 10 equals 20. So, the expression inside the braces becomes 12 + 20 + 1. Adding those up, we get 33. Let’s replace the braces: 2⁵ + 2 × 33 + 1. We’re shedding layers like a mathematical onion!

Now, let's deal with the exponent. We have 2⁵, which means 2 raised to the power of 5. That’s 2 multiplied by itself five times (2 × 2 × 2 × 2 × 2), which equals 32. Let’s substitute that in: 32 + 2 × 33 + 1. The expression is getting simpler and simpler! We are so close now.

Next in line is multiplication. We have 2 × 33, which equals 66. Let's put that back into the expression: 32 + 66 + 1. We’ve handled the multiplication, and now we’re on the home stretch!

Finally, we’re left with addition. We add 32, 66, and 1 together. 32 plus 66 is 98, and adding 1 to that gives us 99. So, the final answer to 2⁵ + 2 × {12 + 2 × [3 × (5 - 2) + 1] + 1} + 1 is 99. You did it! You conquered the mathematical beast! The key takeaway here is that no matter how complex an expression looks, breaking it down step by step, following the order of operations, will always lead you to the solution. Give yourself a pat on the back – you’ve earned it!

1.65: Evaluating Polynomials

a) P = 2x³ + 3x² + 5x + 1 when x = 1

Let's dive into the world of polynomials! These mathematical expressions might look a bit intimidating with their mix of variables and exponents, but trust me, they're just puzzles waiting to be solved. In this case, we have a polynomial P, and we're going to find its value when x is equal to 1. Think of it as a treasure hunt, where substituting the value of x is the key to unlocking the final answer. We'll take it step-by-step, just like we always do, and you'll see how straightforward it can be!

First things first, let’s rewrite our polynomial: P = 2x³ + 3x² + 5x + 1. We're given that x = 1, so our mission is to replace every 'x' in the polynomial with the number 1. It’s like swapping pieces in a game – we’re just substituting one value for another. This substitution is the first key step in evaluating any polynomial for a specific value of the variable.

Now, let's make the substitution: P = 2(1)³ + 3(1)² + 5(1) + 1. Notice how we've replaced each 'x' with '1', and we've put parentheses around the '1' to make it clear that we're multiplying. This is a crucial step to avoid confusion and keep our calculations organized. Now we are set to move to the next step in our evaluation journey.

Next up, we need to tackle those exponents. We've got (1)³ and (1)². Remember, any number raised to the power of 1 is just itself, and 1 raised to any power is always 1. So, (1)³ is 1 × 1 × 1, which equals 1, and (1)² is 1 × 1, which also equals 1. Let’s replace these values in our expression: P = 2(1) + 3(1) + 5(1) + 1. See how simplifying the exponents makes the expression much easier to handle?

Now, let's take care of the multiplication. We've got 2 multiplied by 1, which is 2. Then, 3 multiplied by 1, which is 3. And finally, 5 multiplied by 1, which is 5. Let’s substitute these results back into our equation: P = 2 + 3 + 5 + 1. We’re almost there! We've simplified the exponents and multiplications, which means we're in the home stretch.

Finally, we’re left with addition. We add 2, 3, 5, and 1 together. 2 plus 3 is 5, plus another 5 is 10, and adding 1 to that gives us 11. So, P = 11. That's it! We've found the value of the polynomial when x = 1. The key here is to substitute the value of the variable and then follow the order of operations. You've successfully navigated the world of polynomials – great job!

b) P = a² - 2ab + b² when a = 2, b = -1

Alright, let's tackle another exciting polynomial! This time, we've got a polynomial P with two variables, 'a' and 'b'. Our mission is to find the value of P when a = 2 and b = -1. Think of it as a mathematical puzzle where we have two pieces to fit in – the values of 'a' and 'b'. Just like before, we'll take it step-by-step, and you'll see how manageable these expressions can be!

First things first, let’s rewrite our polynomial: P = a² - 2ab + b². We're given that a = 2 and b = -1. Our mission is to replace every 'a' and 'b' in the polynomial with their respective values. It’s like swapping out players in a game – we’re just substituting one value for another. Remember to be extra careful with negative numbers; they can sometimes trip us up if we’re not paying attention.

Now, let's make the substitution: P = (2)² - 2(2)(-1) + (-1)². Notice how we've replaced 'a' with '2' and 'b' with '-1', and we've put parentheses around the numbers to keep things clear. This is especially important when dealing with negative numbers, as it helps us keep track of the signs. We have successfully replaced the variables with values. Next we simplify and get closer to the final result.

Next up, we need to tackle the exponents. We've got (2)² and (-1)². Remember, a number squared means that number multiplied by itself. So, (2)² is 2 × 2, which equals 4. And (-1)² is -1 × -1, which equals 1 (a negative times a negative is a positive!). Let’s substitute these values back into our expression: P = 4 - 2(2)(-1) + 1. See how simplifying the exponents makes things easier to handle?

Now, let's take care of the multiplication. We have -2 multiplied by 2, which is -4. Then, we multiply -4 by -1. Remember, a negative times a negative is a positive, so -4 × -1 equals 4. Let’s substitute this result back into our equation: P = 4 + 4 + 1. We’ve handled the multiplication, and we are almost ready to calculate our final result now.

Finally, we’re left with addition. We add 4, 4, and 1 together. 4 plus 4 is 8, and adding 1 to that gives us 9. So, P = 9. That's it! We've found the value of the polynomial when a = 2 and b = -1. The key here is to carefully substitute the values of the variables, especially when dealing with negative numbers, and then follow the order of operations. You've successfully conquered another polynomial – fantastic job!

By mastering these mathematical expressions, you're building a solid foundation for more advanced concepts. Keep practicing, keep exploring, and remember that every challenge is an opportunity to grow your math skills. You've got this!