Math Challenge: Simplifying Radicals And Finding Geometric Mean

Hey guys! Let's dive into a fun math problem that involves simplifying radicals and calculating the geometric mean. We'll break down each step so you can easily follow along and understand the concepts. This problem is a great exercise for anyone looking to sharpen their algebra skills. We'll work through the simplification of radical expressions and then apply that knowledge to calculate the geometric mean. So, grab your pencils, and let's get started! This problem is all about mastering the manipulation of radical expressions. We are going to simplify expressions containing square roots and then use these simplified forms to calculate the geometric mean. The problem also incorporates a touch of algebraic manipulation, making it a well-rounded exercise. This is a chance to not only practice your simplification skills but also to deepen your understanding of how radicals behave within mathematical expressions. The ability to work with radicals is crucial in many areas of mathematics, from geometry to calculus, so mastering this skill will set you up for success in your studies. In addition, the geometric mean provides a great application to understand how these numbers interact. Let's make sure that by the end of this exercise, you'll feel confident in your ability to simplify radical expressions and calculate geometric means.

Part A: Simplifying the Expression for 'a'

Okay, so the problem starts with two real numbers, a and b. The first step is to tackle a, which is given as: a = (4/sqrt(2) + 6/sqrt(3)) * sqrt(6) - (sqrt(3) - 2*sqrt(2)) * sqrt(24). Our goal here is to simplify this expression and show that it equals 12*sqrt(3). Let's break this down step by step to avoid any confusion. First, let's look at the first part of the expression: (4/sqrt(2) + 6/sqrt(3)) * sqrt(6). To make things easier, we'll rationalize the denominators of the fractions inside the parentheses. Rationalizing means getting rid of the square roots in the denominators. For the first term, 4/sqrt(2), we multiply both the numerator and the denominator by sqrt(2): (4 * sqrt(2)) / (sqrt(2) * sqrt(2)) = 4*sqrt(2)/2 = 2*sqrt(2). For the second term, 6/sqrt(3), we multiply by sqrt(3)/sqrt(3): (6 * sqrt(3)) / (sqrt(3) * sqrt(3)) = 6*sqrt(3)/3 = 2*sqrt(3). Now, we can rewrite the first part of the expression as: (2*sqrt(2) + 2*sqrt(3)) * sqrt(6). Let's distribute the sqrt(6): 2*sqrt(2) * sqrt(6) + 2*sqrt(3) * sqrt(6). Remember that sqrt(a) * sqrt(b) = sqrt(a*b). So, 2*sqrt(2) * sqrt(6) = 2*sqrt(12). And 2*sqrt(3) * sqrt(6) = 2*sqrt(18). We can simplify sqrt(12) as sqrt(4*3) = 2*sqrt(3), and sqrt(18) as sqrt(9*2) = 3*sqrt(2). Therefore, this part becomes 2*(2*sqrt(3)) + 2*(3*sqrt(2)) = 4*sqrt(3) + 6*sqrt(2). Great! We've simplified the first part.

Next, let's deal with the second part of the original expression: (sqrt(3) - 2*sqrt(2)) * sqrt(24). We can simplify sqrt(24) as sqrt(4*6) = 2*sqrt(6). So, this part becomes (sqrt(3) - 2*sqrt(2)) * 2*sqrt(6). Distribute 2*sqrt(6): sqrt(3) * 2*sqrt(6) - 2*sqrt(2) * 2*sqrt(6) = 2*sqrt(18) - 4*sqrt(12). We already know that sqrt(18) = 3*sqrt(2) and sqrt(12) = 2*sqrt(3). Thus, we have 2*(3*sqrt(2)) - 4*(2*sqrt(3)) = 6*sqrt(2) - 8*sqrt(3). Okay, now we put it all together! The original expression, a, becomes: (4*sqrt(3) + 6*sqrt(2)) - (6*sqrt(2) - 8*sqrt(3)). Combine like terms: 4*sqrt(3) + 8*sqrt(3) + 6*sqrt(2) - 6*sqrt(2) = 12*sqrt(3). And there you have it, guys! We've shown that a = 12*sqrt(3). This detailed breakdown makes it easy to understand the steps involved in simplifying this expression. You will find that this methodical approach, of breaking down the problem into manageable steps, is key to success.

Part B: Calculating the Geometric Mean of 'a' and 'b'

Now, let's move on to Part B, where we're asked to calculate the geometric mean of a and b. We already know that a = 12*sqrt(3). We also need to find the value of b first. The expression for b is given as: b = (3 + sqrt(3))^2 - (sqrt(3) - 3)^2. Recall that the geometric mean of two numbers, x and y, is calculated as sqrt(x*y). So, once we find the value of b, we can calculate the geometric mean of a and b. Let's simplify b. We will use the formula (x + y)^2 = x^2 + 2xy + y^2 and (x - y)^2 = x^2 - 2xy + y^2. For the first term, (3 + sqrt(3))^2 = 3^2 + 2*3*sqrt(3) + (sqrt(3))^2 = 9 + 6*sqrt(3) + 3 = 12 + 6*sqrt(3). For the second term, (sqrt(3) - 3)^2 = (sqrt(3))^2 - 2*3*sqrt(3) + 3^2 = 3 - 6*sqrt(3) + 9 = 12 - 6*sqrt(3). Thus, b = (12 + 6*sqrt(3)) - (12 - 6*sqrt(3)). Simplifying this gives us b = 12 + 6*sqrt(3) - 12 + 6*sqrt(3) = 12*sqrt(3). Interestingly, we see that b is also equal to 12*sqrt(3), just like a. Now that we know a = 12*sqrt(3) and b = 12*sqrt(3), we can calculate the geometric mean. The geometric mean of a and b is sqrt(a*b) = sqrt((12*sqrt(3)) * (12*sqrt(3))). This simplifies to sqrt(144*3) = sqrt(432). To further simplify sqrt(432), we can factor 432: 432 = 144*3. So, sqrt(432) = sqrt(144*3) = sqrt(144) * sqrt(3) = 12*sqrt(3). Therefore, the geometric mean of a and b is 12*sqrt(3). That’s pretty cool, right? In essence, the geometric mean is the same as the values of both a and b in this case. This part of the exercise involves applying your simplification skills from Part A and then understanding how the geometric mean is calculated and applied. Knowing this makes sure that you're well-equipped to tackle similar problems in the future. The ability to manipulate expressions and solve the geometry mean is valuable, and now you have the skills to handle these. With this knowledge, you're building a strong foundation for more advanced math concepts.

In summary:

• We first simplified the radical expression for 'a' to prove it equals 12*sqrt(3). We carefully broke down each step, rationalizing denominators and simplifying radicals along the way. This highlighted the importance of a systematic approach.
• Next, we calculated the value of 'b', which also simplified to 12*sqrt(3). This reinforced our simplification skills.
• Finally, we calculated the geometric mean of 'a' and 'b', which turned out to be 12*sqrt(3). This gave us practice in applying the concept of the geometric mean. This exercise allowed us to connect our ability to simplify with another important mathematical concept.

Hope you enjoyed this mathematical journey! Keep practicing, and you'll become a pro at simplifying radicals and calculating geometric means. Keep up the great work, everyone!