Math Problem Solved: Step-by-Step For 5th Graders

Let's break down this problem and solve it step-by-step, just like you'd do in 5th grade! We'll tackle it piece by piece to make sure it's easy to understand. Understanding order of operations is important, so we'll start with exponents, then move on to multiplication and division, and finally, handle the subtraction. Remember PEMDAS/BODMAS!

Understanding the Problem

The problem we're tackling is this big equation: 3³¹÷3-3²⁹×32⁸⁰÷4-2⁷⁶×43¹²×9×272¹⁰×4×8²×16. It looks intimidating, right? But don't worry, we'll simplify it together. The key here is to remember the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) or BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction). We will address each component by breaking down the steps involved in resolving mathematical expressions by using exponents, multiplication, and division so a fifth-grader can grasp the concept with ease. We will take each component apart and streamline it so that it becomes more manageable to work with. This includes simplifying terms that share the same base and following the order of operations to guarantee precision. This strategy enables us to move methodically through the problem and arrive at the solution in a clear and structured way. We will pay close attention to every aspect and double-check our calculations to make sure we've answered the question in the most accurate way possible. The goal is to not only get to the right answer but also to learn the process. By breaking down the problem into smaller steps, we can better understand the logic and skills needed to tackle difficult math problems. This method builds confidence and helps with math problems later on. Math doesn't have to be scary! With a little patience and focus, we can solve even the toughest-looking problems.

Step-by-Step Solution

Okay, let's dive in! First, we'll rewrite the problem to make it a bit clearer:

3³¹ ÷ 3 - 3²⁹ × 32⁸⁰ ÷ 4 - 2⁷⁶ × 43¹² × 9 × 272¹⁰ × 4 × 8² × 16

Part 1: Simplifying 3³¹ ÷ 3

• Remember that dividing exponents with the same base means we subtract the powers. So, 3³¹ ÷ 3 is the same as 3³¹ ÷ 3¹. That means we do 31 - 1. This is a fundamental rule of exponents, and it's super helpful in simplifying expressions.
• 3³¹⁻¹ = 3³⁰. Guys, we've already simplified the first part! Isn't that cool?

Part 2: Simplifying 3²⁹ × 32⁸⁰ ÷ 4

• First, let's deal with 3²⁹ × 32⁸⁰. Uh oh, these don't have the same base! We can't directly combine them yet. This is where we'll need to think about breaking down the numbers or looking for alternative simplification strategies. Perhaps there's a trick or a different way to approach it. Let’s park this for a bit and see if we can simplify other parts first. Sometimes, solving a big problem is like solving a puzzle – you might need to try different pieces to see what fits.
• Next up is dividing by 4. Right now, we can't easily divide the expression by 4 because we haven't simplified the multiplication part fully. Let's keep this in mind and come back to it once we've simplified the 3²⁹ × 32⁸⁰ part (if possible).

Part 3: Simplifying 2⁷⁶ × 43¹² × 9 × 272¹⁰ × 4 × 8² × 16

This part looks like a monster, but we can tame it! We need to rewrite everything in terms of their prime bases (2 and 3). This is a common strategy when dealing with complex expressions involving multiplication and exponents. By expressing each number as a product of its prime factors raised to powers, we can more easily combine like terms and simplify the entire expression. This technique helps us see the underlying structure of the numbers and how they relate to each other. Here’s how we’ll do it:

• 2⁷⁶: This is already in its simplest form with a base of 2. Great start! We don't need to do anything to this one for now. It's ready to be combined with other terms that have a base of 2.
• 4: 4 is 2², because 2 times 2 equals 4. Easy peasy! We’ve now expressed 4 in terms of its prime base, which is 2.
• 9: 9 is 3², because 3 times 3 equals 9. Another one down! We now have 9 represented in terms of its prime base, 3. This will help us combine it with other terms involving 3.
• 16: 16 is 2⁴ (2 × 2 × 2 × 2 = 16). Awesome! We've expressed 16 as a power of 2, which is its prime base.
• 8²: 8 is 2³, so 8² is (2³)² which equals 2⁶. Remember the rule: (a^m)n = a^(mn)*. This is a crucial rule when dealing with exponents raised to other exponents. By applying this rule, we can simplify complex expressions and make them easier to work with.
• 272¹⁰: 27 is 3³, so 272¹⁰ is (3³)²¹⁰ which equals 3⁶³⁰. Wow, that's a big exponent, but we're handling it like pros! We're methodically breaking down each component, no matter how large the numbers get. This shows the power of our systematic approach.
• 43¹²: 43 is already a prime number! So 43¹² stays as it is. Sometimes, things are simpler than they look! It's good to recognize prime numbers early on, as they can't be simplified further.

Now we substitute these back into the expression:

2⁷⁶ × 43¹² × 3² × 3⁶³⁰ × 2² × 2⁶ × 2⁴

Let's regroup and combine the terms with the same base. This is where things get satisfying, like putting the right pieces of a puzzle together! We'll combine all the 2s together and all the 3s together:

(2⁷⁶ × 2² × 2⁶ × 2⁴) × 3² × 3⁶³⁰ × 43¹²

Now, let's add the exponents for the same bases:

• For 2: 76 + 2 + 6 + 4 = 88. So we have 2⁸⁸.
• For 3: 2 + 630 = 632. So we have 3⁶³².

Our simplified expression for this part is:

2⁸⁸ × 3⁶³² × 43¹²

Putting It All Together (So Far) and Pausing

Okay, deep breaths! We've simplified a lot! Our original problem now looks like this:

3³⁰ - (3²⁹ × 32⁸⁰ ÷ 4) - (2⁸⁸ × 3⁶³² × 43¹²)

Phew! We've made some serious progress. However, the term 3²⁹ × 32⁸⁰ ÷ 4 is still tricky, and the final subtraction will likely result in a massive number that might not be practical for a 5th-grade solution. It seems there may be a typo in the original problem or it's designed to illustrate the complexity of large number arithmetic rather than find a simple numerical answer. The sheer scale of the numbers suggests that there might be an error or an element that requires tools or computation outside the scope of standard fifth-grade math.

Given the complexity and scale of the remaining calculation, it’s good to pause and double-check the original problem. It's possible a small error in the original equation has led to this complexity. If we assume the 32⁸⁰ is actually 3⁸⁰, then we can show how it simplifies significantly

Let’s revisit Part 2 with the possible correction: Simplifying 3²⁹ × 3⁸⁰ ÷ 4

If we consider the problem meant 3⁸⁰ instead of 32⁸⁰, this section simplifies nicely:

1. Combine the exponents with the same base: 3²⁹ × 3⁸⁰ becomes 3²⁹⁺⁸⁰ = 3¹⁰⁹.
2. Divide by 4: 3¹⁰⁹ ÷ 4. Since 4 is 2², we cannot simplify this division further as 3¹⁰⁹ does not have a factor of 2. We'll keep it as 3¹⁰⁹ ÷ 4.

Recalculating with the Corrected Value

Now, if we use 3¹⁰⁹ ÷ 4 in our overall equation, here’s what it looks like:

3³⁰ - (3¹⁰⁹ ÷ 4) - (2⁸⁸ × 3⁶³² × 43¹²)

Observing the Result

Even with the correction of 3⁸⁰ instead of 32⁸⁰, the expression remains incredibly large and complex. The challenge now involves subtracting very large numbers, which realistically goes beyond the scope of a typical 5th-grade math problem. This suggests that the goal might be to understand the simplification process rather than to compute a final numerical answer.

Conclusion

So, guys, we've gone super far in simplifying this problem! We used exponent rules, broke down numbers into their prime factors, and combined like terms. We identified a possible correction to the original problem (3⁸⁰ instead of 32⁸⁰). Even with that fix, the problem leads to calculations that are beyond what's typically done in 5th grade. The key takeaway here isn't necessarily the final answer, but the process we used to simplify the expression. We've shown how to tackle a complex problem by breaking it down into smaller, manageable parts. You've learned about exponent rules, prime factorization, and the order of operations. These skills are super important for future math challenges. You rock!