Simplify Expressions With Ease A Step-by-Step Guide

Hey guys! Today, we're diving into a math problem that looks a bit intimidating at first glance, but trust me, it's totally manageable once we break it down. We're going to simplify the expression: (9 - (2/3)√9)² + (1 - 6)². Don't worry, we'll take it one step at a time, and by the end, you'll be a pro at simplifying expressions like this. So, grab your pencils and let's get started!

Understanding the Order of Operations

Before we jump into the nitty-gritty, it's crucial to remember our good old friend, the order of operations – often remembered by the acronym PEMDAS/BODMAS. This acronym is our roadmap for simplifying mathematical expressions, ensuring we tackle operations in the correct sequence. Let's break it down:

• Parentheses (or Brackets):
  • This is our starting point. We simplify everything inside parentheses or brackets before moving on to other operations. Think of it as tidying up the smaller rooms in our mathematical house before tackling the whole building.
• Exponents (or Orders):
  • Next up are exponents or orders, which involve raising numbers to a power. For instance, 2³ (2 cubed) or 5² (5 squared). We handle these after the parentheses are sorted.
• Multiplication and Division:
  • These operations share the same level of priority. We perform them from left to right, just like reading a sentence. If multiplication comes before division as we read the expression, we do the multiplication first, and vice versa.
• Addition and Subtraction:
  • Last but not least, we have addition and subtraction. Similar to multiplication and division, they have equal priority and are performed from left to right.

Why is this order so important, you ask? Well, imagine cooking a complex dish without following the recipe's instructions. You might end up with a culinary disaster! The order of operations is like the recipe for mathematics. It ensures everyone arrives at the same correct answer, no matter who's doing the calculations. Without it, we'd have mathematical chaos!

For our expression, (9 - (2/3)√9)² + (1 - 6)², PEMDAS tells us we need to first simplify inside the parentheses, then deal with the exponent, and finally handle the addition. Keep this in mind as we move forward – it's our guiding principle. Now that we've refreshed our understanding of the order of operations, we're well-equipped to dissect our expression and simplify it with confidence. Let's dive in and see how this knowledge helps us conquer the problem!

Step 1: Simplifying Inside the Parentheses

Okay, guys, let's put our PEMDAS knowledge to work! Our expression is (9 - (2/3)√9)² + (1 - 6)², and the first thing we need to tackle is what's hiding inside those parentheses. Remember, parentheses are like the rooms we need to tidy up before we can clean the whole house.

Let's start with the first set of parentheses: (9 - (2/3)√9). We've got a few operations going on in here, so we need to be strategic. Inside these parentheses, we have subtraction, multiplication, and a square root. According to PEMDAS, we handle the square root first.

The square root of 9 (√9) is 3. So, we can replace √9 with 3, making our expression inside the first parentheses look like this: (9 - (2/3) * 3). Now we have subtraction and multiplication. PEMDAS tells us multiplication comes before subtraction, so let's multiply (2/3) by 3.

(2/3) * 3 is the same as (2 * 3) / 3, which equals 6/3. And 6/3 simplifies to 2. So, now our first set of parentheses looks even simpler: (9 - 2). Finally, we can subtract 2 from 9, which gives us 7. So, the first set of parentheses simplifies down to a neat and tidy 7.

Now, let's move on to the second set of parentheses: (1 - 6). This one's much simpler! We just need to subtract 6 from 1. 1 - 6 equals -5. So, the second set of parentheses simplifies to -5.

Phew! We've successfully simplified everything inside the parentheses. Our original expression, (9 - (2/3)√9)² + (1 - 6)², now looks like this: 7² + (-5)². See how much cleaner it is already? By methodically working through the parentheses, we've made the expression much easier to handle. Now, we're ready to move on to the next step: dealing with those exponents. Stay tuned, we're making great progress!

Step 2: Evaluating the Exponents

Alright, team, we've conquered the parentheses, and now it's time to tackle those exponents! Our expression is currently 7² + (-5)². Remember, an exponent tells us how many times to multiply a number by itself. So, 7² means 7 multiplied by itself, and (-5)² means -5 multiplied by itself.

Let's start with 7². This is simply 7 * 7, which equals 49. Easy peasy!

Now, let's move on to (-5)². This means -5 multiplied by -5. Remember the rules of multiplying negative numbers: a negative times a negative equals a positive. So, -5 * -5 equals 25. It's important to keep track of those signs, guys! They can make a big difference in the final answer.

So, we've figured out that 7² is 49 and (-5)² is 25. We can now replace those exponents in our expression. Our expression, which was 7² + (-5)², now becomes 49 + 25. See how we're gradually making the expression simpler and simpler? By tackling the exponents, we've reduced the complexity and brought ourselves closer to the final answer. We're on a roll!

We've successfully evaluated the exponents, and we're left with a straightforward addition problem. This is the home stretch, guys! We're almost there. Now, let's move on to the final step and add those numbers together to get our simplified answer.

Step 3: Performing the Addition

Okay, folks, we've reached the final stage of our simplification journey! We've conquered the parentheses, tamed the exponents, and now we're left with a simple addition problem. Our expression is 49 + 25. This is the home stretch, the final sprint to the finish line!

Adding 49 and 25 is pretty straightforward. You can do it in your head, on paper, or even use a calculator if you like. But let's break it down just to be crystal clear. You can think of it as 49 + 20 + 5. 49 + 20 is 69, and then adding 5 to 69 gives us 74. So, 49 + 25 equals 74.

And there you have it! We've successfully simplified the entire expression. Our original, somewhat intimidating expression, (9 - (2/3)√9)² + (1 - 6)², has been transformed into a single, neat number: 74. Give yourselves a pat on the back, guys! You've done an awesome job!

We started with a complex expression, but by methodically following the order of operations (PEMDAS/BODMAS) and breaking the problem down into smaller, manageable steps, we were able to arrive at the solution with confidence. This is the beauty of mathematics – taking something complicated and making it simple through logical steps.

Final Answer and Key Takeaways

So, our final answer is 74. We've successfully simplified the expression (9 - (2/3)√9)² + (1 - 6)².

But more than just getting the answer, let's reflect on the key takeaways from this exercise. What did we learn along the way that we can apply to other math problems?

• The Power of PEMDAS/BODMAS: We saw firsthand how crucial the order of operations is. It's the foundation for simplifying any mathematical expression. Remember those parentheses, exponents, multiplication and division, and finally, addition and subtraction. They're our roadmap to success.
• Break It Down: Complex problems can feel overwhelming, but by breaking them down into smaller steps, they become much more manageable. We tackled the parentheses first, then the exponents, and finally the addition. This step-by-step approach is a powerful problem-solving strategy.
• Attention to Detail: We saw how important it is to pay attention to the details, especially the signs. A simple negative sign in the wrong place can throw off the entire calculation. Double-checking your work is always a good idea.
• Practice Makes Perfect: Like any skill, simplifying expressions takes practice. The more you practice, the more comfortable and confident you'll become. Don't be afraid to tackle challenging problems – they're opportunities to learn and grow.

So, there you have it! We've not only simplified a complex expression, but we've also reinforced some fundamental mathematical principles. Keep these takeaways in mind as you continue your mathematical adventures. You've got this, guys!

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Simplify (9-(2/3)√9)²+(1-6)²