Math Challenge: Solving Complex Expressions Step-by-Step

Hey guys! Ready to dive into some math problems? We're going to tackle two expressions that might look a little intimidating at first, but trust me, we'll break them down step by step and make them super easy to understand. This is all about practicing your order of operations, exponents, and a little bit of arithmetic. So, grab your pencils, and let's get started! We will address the first part of the problem and then the second one, covering all the steps to arrive at the correct answer.

Part A: Breaking Down the First Expression

Let's start with the first expression: 5 * [5 + 5 * (5^2 - 5^5 : 5^4)] + 3 * 5^2. This one looks like a mouthful, but we'll break it down bit by bit. Remember the order of operations: Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right) – often remembered by the acronym PEMDAS or BODMAS. This helps us solve expressions in the correct order to avoid confusion and ensure we get the right answer. We begin by addressing the innermost set of parentheses and working our way outwards.

First things first, let's look inside the parentheses. We have 5^2 - 5^5 : 5^4. Within these parentheses, we need to handle the exponents and division before we can do the subtraction. Let's start with the exponents. 5^2 means 5 multiplied by itself twice (5 * 5), which equals 25. Next, 5^5 means 5 multiplied by itself five times, and 5^4 means 5 multiplied by itself four times. This can be written as 5^5 : 5^4. When dividing exponents with the same base, you subtract the powers. So, 5^5 : 5^4 is the same as 5^(5-4), which simplifies to 5^1 or simply 5. So now our expression inside the parentheses becomes 25 - 5, which equals 20. Therefore, 5^2 - 5^5 : 5^4 = 20. Now we can substitute this value back into the original equation. So far, we've simplified the innermost parentheses to 20. The expression now looks like this: 5 * [5 + 5 * 20] + 3 * 5^2. Amazing, right? The next step involves multiplication inside the brackets. The order of operations tells us that we need to multiply 5 * 20 first. This is equal to 100. So we replace this inside the brackets and get 5 + 100. Adding these two numbers together gives us 105. Now the equation looks like this: 5 * [105] + 3 * 5^2. We're getting closer! Next, we address the remaining exponents. We have 3 * 5^2, which means 3 * (5 * 5) = 3 * 25. This is equal to 75. Now the equation is 5 * 105 + 75. Then, we perform the multiplication: 5 * 105 = 525. So, the equation becomes 525 + 75. Finally, we add 525 + 75, which equals 600. So, the answer to the first expression is 600!

Part B: Cracking the Second Expression

Now, let's take a look at the second expression: **(6 + 2^10} 2^8) : 10^5 - 9 * 10**. This one also involves parentheses and exponents. Let's get to it! Again, we begin by working inside the parentheses. Inside the parentheses, we have `2^{10 : 2^8. This involves division with exponents. When dividing exponents with the same base, we subtract the powers. So, 2^{10} : 2^8simplifies to2^(10-8), which equals 2^2. And 2^2is equal to 4. Thus, the equation within the parentheses simplifies to6 + 4, which is 10. Now the expression looks like this: 10 : 10^5 - 9 * 10. The next step is to address the exponents. 10^5` means 10 multiplied by itself five times (10 * 10 * 10 * 10 * 10), which equals 100,000. So now our expression is `10 : 100,000 - 9 * 10`. Next, we do the division and the multiplication. So, first, we divide: `10 : 100,000`, which equals 0.0001. Then, we multiply: `9 * 10 = 90`. Now we have `0.0001 - 90`. Subtracting 90 from 0.0001 gives us -89.9999. So, the answer to the second expression is -89.9999!

Key Takeaways and Tips

Alright, guys, we've made it through both expressions! Here are some key takeaways:

• Order of Operations: Always remember PEMDAS (or BODMAS) – Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). This is the most important rule.
• Break it Down: Don't be afraid to break down complex expressions into smaller, more manageable steps. This makes it easier to avoid mistakes.
• Exponents: Remember that exponents represent repeated multiplication. Be careful when calculating these.
• Practice Makes Perfect: The more you practice, the better you'll get at solving these types of problems. Try working through similar examples on your own.
• Double-Check: Always double-check your work, especially when dealing with multiple steps and calculations. This can help you catch any errors before you get to the final answer.

By following these steps, you'll be well on your way to mastering these types of math problems. The critical aspect is understanding the sequence of operations and consistently applying them. If you find a problem difficult, remember to return to the basic principles. With practice and a solid understanding of the rules, you'll be solving even the most complex equations with confidence. Keep up the great work, and don't hesitate to ask for help if you need it. You've got this! Keep practicing, and you'll become a math whiz in no time! Remember, math is all about practice, so keep at it, and you'll improve with every problem you solve.

Conclusion: You've Got This!

So there you have it! We've successfully solved two complex math expressions. Hopefully, this step-by-step breakdown helped you understand the process and boosted your confidence in tackling similar problems. Math can be challenging, but with practice and the right approach, anyone can master it. Keep practicing, and don't be afraid to challenge yourself with more complex equations. You've totally got this! Go forth, and conquer those math problems, and feel free to ask if anything is unclear. We're all in this together!