Solving The Number's Inverse: A Math Problem Explained

Hey guys! Let's dive into a cool math problem that involves finding the inverse of a number. This kind of problem often pops up in math exams and it's super important to understand the basics. The problem we're going to solve is: "98. 2. The inverse of the number a = 1/√(5-3√3)^2 - 2/√(2√3-4)^2 is: A. -√3; B. √3; C. √3-1; D. √3+1." Sounds a bit intimidating, right? Don't worry, we'll break it down step by step and make it easy to follow. Our main goal is to find the inverse of the given expression, and to do that, we'll need to simplify the expression first. This involves dealing with square roots, squares, and fractions. Ready to get started? Let's go!

Understanding the Basics: Inverses and Simplifying Expressions

Alright, before we jump into the problem, let's quickly recap what an inverse is. In simple terms, the inverse of a number is what you multiply it by to get 1. For instance, the inverse of 2 is 1/2 because 2 * (1/2) = 1. The inverse of -3 is -1/3 because -3 * (-1/3) = 1. See? It's all about getting back to 1. Now, our problem is a bit more complex since we're dealing with an expression rather than a single number. We'll have to simplify the expression first to find the value of 'a,' and then determine its inverse. This involves dealing with squares and square roots. Remember that the square root of a number squared is the absolute value of that number. For example, √(x^2) = |x|. This is a crucial concept. So, if we have a term like √(5-3√3)^2, we'll take the absolute value of (5-3√3). Similarly, for √(2√3-4)^2, we'll take the absolute value of (2√3-4). The absolute value ensures we get a positive result, which will be essential in our calculations. Understanding how to handle these absolute values and simplify the expressions will make finding the inverse much simpler, so it's a good time to get comfortable with the concept before we move on.

Step-by-Step Breakdown: Simplifying the Expression

Now, let's get our hands dirty and simplify the given expression step by step. This is where the real fun begins! Remember, our expression is: a = 1/√(5-3√3)^2 - 2/√(2√3-4)^2. We'll start by addressing the square roots and squares.

1. Simplify √(5-3√3)^2: As mentioned earlier, √(x^2) = |x|. So, √(5-3√3)^2 = |5-3√3|. To determine the absolute value, we need to know whether 5-3√3 is positive or negative. Since √3 is approximately 1.73, then 3√3 ≈ 5.19. Thus, 5 - 3√3 is negative. Therefore, |5-3√3| = -(5-3√3) = 3√3 - 5.
2. Simplify √(2√3-4)^2: Similarly, √(2√3-4)^2 = |2√3-4|. Using the approximation, 2√3 ≈ 3.46. Thus, 2√3 - 4 is also negative, which means |2√3-4| = -(2√3-4) = 4 - 2√3.

Now, let's rewrite our original expression, replacing the simplified square roots:

a = 1 / (3√3 - 5) - 2 / (4 - 2√3)

Rationalizing the Denominators

When we have expressions with square roots in the denominator, it's generally good practice to rationalize the denominator. This makes the expression easier to work with. Here’s how we do it:

1. Rationalize 1 / (3√3 - 5): To rationalize this, we multiply both the numerator and denominator by the conjugate of the denominator, which is (3√3 + 5). So, (1 / (3√3 - 5)) * ((3√3 + 5) / (3√3 + 5)) = (3√3 + 5) / ((3√3)^2 - 5^2) = (3√3 + 5) / (27 - 25) = (3√3 + 5) / 2.
2. Rationalize 2 / (4 - 2√3): The conjugate of (4 - 2√3) is (4 + 2√3). Multiplying both numerator and denominator by the conjugate gives us: (2 / (4 - 2√3)) * ((4 + 2√3) / (4 + 2√3)) = (8 + 4√3) / (16 - 12) = (8 + 4√3) / 4 = 2 + √3.

Now, our expression becomes:

a = (3√3 + 5) / 2 - (2 + √3)

Combining Terms and Finding 'a'

Now that we have rationalized the denominators, it's time to combine the terms and find the value of 'a'.

a = (3√3 + 5) / 2 - 2 - √3

Let’s simplify this further.

a = (3√3 / 2 + 5 / 2) - 2 - √3

To combine the √3 terms, we can rewrite the expression and find common denominators.

a = (3√3 / 2 - 2√3 / 2) + 5 / 2 - 4 / 2

a = √3 / 2 + 1 / 2

So, 'a' = (√3 + 1) / 2. This is the value of 'a'. Don't worry; we are getting closer to the solution.

Calculating the Inverse: The Final Steps

Alright, we've done all the hard work to simplify the original expression and find the value of 'a'. The final step is to determine the inverse of 'a'. Remember, the inverse is a number that, when multiplied by 'a,' results in 1. Because 'a' = (√3 + 1) / 2, its inverse is the reciprocal of this value. So, we'll simply flip the fraction.

Finding the Inverse of a

To find the inverse of a, we simply take the reciprocal of the value we found for 'a.'

Inverse of a = 2 / (√3 + 1)

However, it's customary to rationalize the denominator of the inverse as well. To rationalize 2 / (√3 + 1), multiply the numerator and denominator by the conjugate of the denominator (√3 - 1):

(2 / (√3 + 1)) * ((√3 - 1) / (√3 - 1)) = (2(√3 - 1)) / (3 - 1) = (2√3 - 2) / 2 = √3 - 1.

Therefore, the inverse of 'a' is √3 - 1. We finally have our answer!

Conclusion: The Answer and What We Learned

So, after all that work, the correct answer is C. √3 - 1. Congratulations, you made it! This problem showed us how to deal with square roots, absolute values, rationalizing denominators, and finding inverses. Remember, practice is key! The more you work through these kinds of problems, the easier they become. Keep practicing and exploring new mathematical concepts. That's the key to mastering math! This journey demonstrated how fundamental algebra concepts fit together to solve seemingly complicated problems.

Summary of Key Steps

• Simplify Square Roots: Use absolute values to handle squared square roots. This avoids common pitfalls. This step is usually overlooked, but it is super crucial for accuracy.
• Rationalize Denominators: Make expressions easier to work with. If you are not familiar with conjugates, this is the time to master it.
• Combine Terms: Simplify and isolate the variable.
• Find the Inverse: Take the reciprocal of 'a' and rationalize the denominator.

Keep practicing, and good luck with your math studies! You got this!