Simplifying Math Problems: A Deep Dive Into Number Theory

Hey guys, let's dive into a cool math problem! We're gonna break down some expressions and see how we can simplify them. This stuff is super helpful, not just for your math class, but also for developing your problem-solving skills overall. We'll tackle the problem step-by-step, making sure everything is crystal clear. Ready to get started? Let's go!

Understanding the Problem: Breaking Down the Expressions

Alright, first things first, let's understand what we're dealing with. We've got two numbers, a and b, defined by some mathematical expressions. Our goal is to simplify these expressions and demonstrate a specific relationship. This is all about simplifying expressions, which is a fundamental skill in algebra and number theory. The ability to simplify and manipulate these expressions is going to make tougher problems way easier down the line. So let's start with the first number, a. It's given by: a = 9/√3 - 3/2 √(1+1/3) + √((14/5) * (20/7)).

See, we are starting with a and as you can see it involves a square root and some fractions. Then we have a slightly more complex expression for b: b = √12 - √8. The problem asks us to show that a simplifies to a particular form, and we are going to see how these expressions relate to each other. We're going to use several techniques here: rationalizing denominators, simplifying square roots, and combining like terms. Remember, the trick to simplifying expressions is breaking them down into manageable parts and performing operations carefully. Math is all about precision, so take your time and double-check each step. That way, it will make your learning process easier. It is like building a Lego set. You have to put each piece in the correct spot. By doing this, we will gain a deeper understanding of how mathematical expressions can be manipulated and simplified. Are you ready to start? Let's go!

Step-by-Step Simplification of Number 'a'

Let's take the first part of number a: 9/√3. The first thing we wanna do here is rationalize the denominator. This means getting rid of the square root in the denominator. To do this, we multiply both the numerator and the denominator by √3. So, we get (9 * √3) / (√3 * √3), which simplifies to 9√3 / 3. Now, we can divide 9 by 3, giving us 3√3. Voila! The first part of a has been simplified! That's how it works guys. Now for the second part. The second part of a is -3/2 √(1+1/3). Let's focus on what's inside the square root first. 1 + 1/3 is the same as 3/3 + 1/3, which equals 4/3. So now, we have -3/2 √(4/3). We can simplify the square root of 4 to 2, so we get -3/2 * (2/√3). The 2's cancel out, leaving us with -3/√3. Again, we rationalize the denominator by multiplying by √3/√3, giving us -3√3 / 3. This simplifies to -√3. Now, we're cooking with fire! We now have -√3 as the second part. Let's move to the last part of the expression for a: √((14/5) * (20/7)). First, let's simplify what's inside the square root. We have (14/5) * (20/7). The 14 and 7 can be simplified to 2, and the 20 and 5 can be simplified to 4. Then, 2 times 4 = 8. It means we are left with √8. Let's now simplify the √8. √8 can be broken down into √(4*2), which is equal to 2√2. Great, we've simplified the last part of a. Now that we have simplified each part of the expression for a, we can combine them.

Putting It All Together: Solving for 'a'

So far, we have simplified the expression for a. Now, let's recap what we've found and put it all together. We started with a = 9/√3 - 3/2 √(1+1/3) + √((14/5) * (20/7)). We've simplified each part separately. The first part, 9/√3, became 3√3. The second part, -3/2 √(1+1/3), became -√3. And the third part, √((14/5) * (20/7)), became 2√2. So, the expression for a now looks like this: a = 3√3 - √3 + 2√2. Now, we can combine like terms. We have 3√3 - √3, which equals 2√3. So, a = 2√3 + 2√2. We've successfully simplified the expression for a and shown that a = 2√3 + 2√2. Boom! We proved it! Guys, that wasn't so hard, right? Now, on to part two.

Analyzing Number 'b' and the Final Verification

Now let's jump into number b. b is defined as √12 - √8. We need to simplify it to see if it is related to our value of a. First, let's simplify √12. We can break this down into √(43), which simplifies to 2√3. Got it? Now, let's simplify √8. We can break this down into √(42), which simplifies to 2√2. So, b = 2√3 - 2√2. It seems that we were asked to demonstrate that a = 2√3 + 2√2. We've already done that, haven't we? Now that we've simplified b and found the value of a, the problem doesn't ask us to establish a direct relationship between a and b but focuses on the simplification of a. We have simplified a, and have proved that a = 2√3 + 2√2. We're done! We did it! We broke down the expressions, simplified them step-by-step, and verified our result. This is how you tackle problems with square roots and fractions. Remember, practice makes perfect. So keep working on problems like these, and you'll become a pro in no time. Keep up the great work!