Solving The Math Problem: √(3*36) + 7 - 8√0.25 + √(20²-16²)
Hey guys! Today, we're diving into a fun math problem that involves square roots, multiplication, subtraction, and a bit of exponentiation. Let's break it down step-by-step so it’s super easy to follow. We will meticulously solve the expression √(3 * 36) + 7 - 8 * √0.25 + √(20² - 16²). Our main goal is not just to find the answer, but to understand the process, so grab your calculators (or your brains!) and let’s get started!
Understanding the Order of Operations
Before we jump into the nitty-gritty, it’s essential to remember the order of operations, often remembered by the acronym PEMDAS (or BODMAS in some parts of the world):
- Parentheses (or Brackets)
- Exponents (or Orders)
- Multiplication and Division (from left to right)
- Addition and Subtraction (from left to right)
This order ensures that we tackle the problem in the correct sequence, leading us to the right solution. Ignoring this order can lead to a completely different answer, which we definitely want to avoid!
Breaking Down the Expression
Our expression is: √(3 * 36) + 7 - 8 * √0.25 + √(20² - 16²). To solve this, we’ll take it one piece at a time, following PEMDAS. This approach not only helps in simplifying the problem but also makes it less daunting. Math can be fun when you break it down, trust me!
Step-by-Step Solution
1. Simplify Inside the Square Roots
First, let’s tackle the square roots. We have three square root terms in our expression:
- √(3 * 36)
- √0.25
- √(20² - 16²)
1.1. √(3 * 36)
We start by multiplying inside the square root:
3 * 36 = 108
So, now we have √108. We'll deal with this after simplifying the other square roots first.
1.2. √0.25
This one’s a bit more straightforward. The square root of 0.25 is 0.5 because 0.5 * 0.5 = 0.25. So:
√0.25 = 0.5
1.3. √(20² - 16²)
Here, we need to handle the exponents first:
20² = 20 * 20 = 400
16² = 16 * 16 = 256
Now, subtract:
400 - 256 = 144
So, we have √144. The square root of 144 is 12, because 12 * 12 = 144. Therefore:
√(20² - 16²) = √144 = 12
2. Rewrite the Expression
Now that we’ve simplified the square roots, let’s rewrite our expression with the simplified values:
√108 + 7 - 8 * 0.5 + 12
3. Simplify the Remaining Square Root
We still have √108 to deal with. To simplify this, we need to find the largest perfect square that divides 108. The largest perfect square factor of 108 is 36 (since 36 * 3 = 108). So, we can rewrite √108 as:
√108 = √(36 * 3) = √36 * √3 = 6√3
Now our expression looks like this:
6√3 + 7 - 8 * 0.5 + 12
4. Perform Multiplication
Next up is the multiplication:
8 * 0.5 = 4
So, our expression becomes:
6√3 + 7 - 4 + 12
5. Perform Addition and Subtraction
Now we’re in the home stretch! Let’s do the addition and subtraction from left to right:
7 - 4 = 3
So, we have:
6√3 + 3 + 12
Now add 3 and 12:
3 + 12 = 15
Our expression simplifies to:
6√3 + 15
6. Final Result
So, the final simplified form of the expression is:
6√3 + 15
This is the exact form of our answer. If you need a decimal approximation, you can use a calculator to find the value of √3 (approximately 1.732) and then compute the result:
6 * 1.732 + 15 ≈ 10.392 + 15 ≈ 25.392
Therefore, the approximate value of the expression is 25.392.
Key Concepts Used
Let’s quickly recap the key math concepts we used to solve this problem. Understanding these concepts is super important for tackling similar problems in the future.
1. Order of Operations (PEMDAS/BODMAS)
We followed the correct order of operations to ensure we solved the expression accurately. Remember, it's Parentheses/Brackets, Exponents/Orders, Multiplication and Division, and finally, Addition and Subtraction.
2. Square Roots
We simplified expressions involving square roots, understanding how to break down numbers into their factors to find perfect squares. This is a crucial skill when dealing with square roots.
3. Exponents
We calculated exponents (like 20² and 16²) by multiplying the base by itself the number of times indicated by the exponent. It’s basic but essential!
4. Simplifying Radicals
We simplified √108 by finding the largest perfect square factor (36) and breaking it down into 6√3. Simplifying radicals helps in getting the expression into its simplest form.
Why is this Important?
You might be wondering, “Why do I need to know this stuff?” Well, these mathematical skills are crucial not just for math class but also for many real-world situations. From calculating areas and volumes to understanding financial investments, these concepts pop up everywhere. Plus, math is like a mental workout – it sharpens your brain and helps you think critically!
Practice Makes Perfect
If you found this tricky, don’t worry! Math becomes easier with practice. Try solving similar problems to build your confidence. You can even make up your own expressions and challenge yourself. The more you practice, the more comfortable you’ll become with these concepts.
Conclusion
So, there you have it! We successfully solved the expression √(3 * 36) + 7 - 8 * √0.25 + √(20² - 16²) by breaking it down step-by-step, using the order of operations, and simplifying square roots. Remember, math is all about understanding the process, not just memorizing formulas. Keep practicing, and you’ll become a math whiz in no time!
I hope this explanation was helpful and fun. Keep exploring the world of math, guys! You’ve got this! Math isn't just about numbers; it's about problem-solving, critical thinking, and building a foundation for future learning and real-world applications.