Simplifying Radicals: (2 + 3√5) / (√8 + √2) Explained

Hey math enthusiasts! Today, we're diving into the world of simplifying radicals. We'll tackle the expression (2 + 3√5) / (√8 + √2). Don't worry if it looks a bit intimidating at first; we'll break it down into easy-to-follow steps. By the end, you'll be a pro at simplifying radicals and feeling confident with your math skills. Let's get started, shall we?

Understanding the Problem: Decoding the Radicals

So, what exactly are we dealing with? The expression (2 + 3√5) / (√8 + √2) involves radicals, which are square roots in this case. These can sometimes look a bit messy, but they're just numbers. The key here is to simplify the radicals to make the overall expression easier to work with. Think of it as cleaning up a cluttered room; once you organize everything, it looks much better. Before we jump into the actual simplification, it's important to understand what a radical is. A radical, represented by the symbol √, asks the question, “What number, when multiplied by itself, gives you this value?” For example, √9 asks “What number multiplied by itself equals 9?” The answer, of course, is 3. In our expression, we have √5, √8, and √2. √5 is already in its simplest form because 5 has no perfect square factors other than 1. However, √8 can be simplified. To simplify a radical, we look for perfect square factors within the number under the radical. A perfect square is a number that is the result of squaring an integer (e.g., 4, 9, 16, 25).

Now, let's identify perfect square factors in our radicals. We have √8, which can be rewritten as √(4 * 2). Here, 4 is a perfect square (2 * 2 = 4). This means we can simplify √8 as follows: √8 = √(4 * 2) = √4 * √2 = 2√2. √2 is already in its simplest form because 2 doesn’t have any perfect square factors other than 1. Now that we've covered the basics, let's move on to the step-by-step solution. We will break down the problem into smaller, more manageable chunks.

Step-by-Step Solution: Simplifying the Expression

Alright, let's get our hands dirty and simplify (2 + 3√5) / (√8 + √2). We will go through this step by step. Remember, we already simplified √8 to 2√2. This is our starting point. First, replace √8 with 2√2 in the original expression. So, (√8 + √2) becomes (2√2 + √2). Now our expression is (2 + 3√5) / (2√2 + √2). Next, we can simplify the denominator, (2√2 + √2). Both terms have √2, so we can combine them like terms. Think of it like having 2 apples + 1 apple = 3 apples. In this case, we have 2√2 + 1√2 = 3√2. That turns our expression into (2 + 3√5) / (3√2). Now comes the rationalization step. We need to get rid of the radical in the denominator, which is √2. We do this by multiplying both the numerator and the denominator by the conjugate of the denominator. The conjugate of a binomial (an expression with two terms) is the same expression but with the opposite sign in the middle. However, since we only have one term (3√2), the concept of conjugate doesn't really apply. To rationalize the denominator in this case, we simply multiply both the numerator and denominator by √2. This is equivalent to multiplying the whole fraction by 1, so it doesn't change the value of the expression, just its form. Multiplying the numerator and denominator by √2 gives us:

• Numerator: (2 + 3√5) * √2 = 2√2 + 3√10.
• Denominator: 3√2 * √2 = 3 * 2 = 6.

So our expression now looks like this: (2√2 + 3√10) / 6. We can look for opportunities to simplify further. Here, it does not look like any further simplification is possible. There are no common factors for 2, 3, and 6. Therefore, we can conclude that the expression is already in its simplest form. But, there is an extra step you could perform to present a cleaner form of the answer by splitting the fraction and reducing where possible. Let us walk through it.

Rationalizing the Denominator and Final Simplification

Let’s focus on rationalizing the denominator and obtaining the final simplified form of (2 + 3√5) / (√8 + √2). We already know the first steps of simplifying the radical expressions. We simplified our expression down to (2√2 + 3√10) / 6. Our aim now is to get rid of the radical in the denominator, which, in this case, we already have achieved by following the previous steps. Rationalizing the denominator involves eliminating any radicals from the denominator. This is done by multiplying both the numerator and denominator by a value that will eliminate the radical. Since we have already rationalized, let's look for an alternative simplification. Remember, our expression is currently (2√2 + 3√10) / 6. Let's try splitting the fraction and simplifying further if possible. We can split this fraction into two separate fractions: (2√2 / 6) + (3√10 / 6). Now, we can look at each of the terms separately to see if we can simplify them. In the first term, 2√2 / 6, we can simplify the fraction 2/6 to 1/3. This gives us (1/3)√2, or √2 / 3. In the second term, 3√10 / 6, we can simplify the fraction 3/6 to 1/2. This gives us (1/2)√10, or √10 / 2. Now, we have simplified our expression to (√2 / 3) + (√10 / 2). Although it does look simpler than the original expression, (2 + 3√5) / (√8 + √2), keep in mind that both answers are correct, and both have their place. The final simplified answer, and the most elegant representation of the expression, is (√2 / 3) + (√10 / 2). Thus, the simplified form of (2 + 3√5) / (√8 + √2) is (√2 / 3) + (√10 / 2).

Conclusion: Mastering Radical Simplification

Congrats, guys! You've successfully simplified the radical expression (2 + 3√5) / (√8 + √2)! We started with a seemingly complex problem, but by breaking it down into smaller steps, simplifying radicals, rationalizing the denominator, and simplifying fractions, we arrived at the final, simplified answer: (√2 / 3) + (√10 / 2). This entire process not only involves mathematical operations but also strengthens your logical thinking and problem-solving skills. It helps you appreciate the underlying structure and the ability to manipulate it to arrive at elegant solutions. This is a victory! Keep practicing these techniques and tackling similar problems, and you'll find that simplifying radicals becomes second nature. So, keep exploring, keep learning, and keep those math skills sharp. The journey of mastering math is an ongoing adventure! Keep practicing, and you will master the simplification of radicals like (2 + 3√5) / (√8 + √2).