Math Challenge: Simplifying Radicals & Expressions

Hey guys! Let's dive into some radical simplification problems. We've got a bunch of expressions to break down and a neat little calculation involving nested radicals. Get your pencils ready, because we're about to make math fun! We'll tackle each expression step-by-step, so you can follow along and understand exactly how to simplify these radicals. Let's get started and turn those complicated expressions into something much cleaner and easier to handle.

Calculating 'a' Given a = √(56 + √8)

First, we're given that a = √(56 + √8), and our mission is to calculate the value of 'a'. This involves simplifying the nested radical. The key here is to recognize that we need to simplify √8 first before we can proceed. √8 can be expressed as √(4 * 2), which simplifies to 2√2. So, our expression becomes a = √(56 + 2√2). Now, we need to see if we can express 56 + 2√2 in the form of (x + y)², where x and y involve radicals. This is a bit tricky, but we are not required to further simplify it, so we keep a = √(56 + 2√2). The important thing is understanding how to simplify the initial radical and recognize the form we're aiming for. This forms a foundation for dealing with more complex nested radicals later on. Remember, the goal isn't always to find a simple numerical answer, but rather to simplify the expression as much as possible and understand the underlying mathematical principles. Keep practicing, and you'll become a pro at simplifying radicals in no time!

Simplifying Radical Expressions

Now, let's move on to the main course: simplifying those radical expressions! We will use the distributive property and simplify each term to its simplest radical form. We'll combine like terms where possible to get the final simplified expression. Remember, simplifying radicals is all about finding perfect square factors within the radical and pulling them out.

a) √3 (√15 - √6)

In this expression, we will distribute √3 across the terms inside the parenthesis. This gives us √3 * √15 - √3 * √6. Now, let's simplify each term separately. √3 * √15 can be written as √(3 * 15) = √45. We can simplify √45 by finding its perfect square factor, which is 9. So, √45 = √(9 * 5) = 3√5. Next, let's simplify √3 * √6, which can be written as √(3 * 6) = √18. Again, we find the perfect square factor, which is 9. So, √18 = √(9 * 2) = 3√2. Putting it all together, we have 3√5 - 3√2. Since √5 and √2 are different radicals, we cannot combine them further. Therefore, the simplified expression is 3√5 - 3√2. This entire process highlights the importance of recognizing perfect square factors and applying the distributive property correctly. This is essential for simplifying any radical expressions you come across.

b) (√75 - √60 + 4√54) : √3

Here, we have (√75 - √60 + 4√54) : √3. It's the same as (√75 - √60 + 4√54) / √3. Let's first simplify the radicals inside the parenthesis. √75 can be simplified as √(25 * 3) = 5√3. √60 can be simplified as √(4 * 15) = 2√15. √54 can be simplified as √(9 * 6) = 3√6. Substituting these back into the expression, we get (5√3 - 2√15 + 4 * 3√6) / √3, which simplifies to (5√3 - 2√15 + 12√6) / √3. Now, we divide each term by √3. (5√3 / √3) - (2√15 / √3) + (12√6 / √3) = 5 - 2√(15/3) + 12√(6/3) = 5 - 2√5 + 12√2. So, the simplified expression is 5 - 2√5 + 12√2. Remember, division by a radical is equivalent to rationalizing the denominator, making sure we handle each term meticulously.

c) 3√5 (√10 + √5 - 2)

In this case, we distribute 3√5 across the terms in the parenthesis: 3√5 * √10 + 3√5 * √5 - 3√5 * 2. Let's simplify each term. 3√5 * √10 = 3√(5 * 10) = 3√50 = 3√(25 * 2) = 3 * 5√2 = 15√2. 3√5 * √5 = 3 * 5 = 15. 3√5 * 2 = 6√5. Putting it all together, we get 15√2 + 15 - 6√5. Since the radicals are different, we cannot combine any further. So, the simplified expression is 15√2 + 15 - 6√5. Distributing and then simplifying is the key to these problems!

d) (√48 - 2√135) : √12

We can rewrite this as (√48 - 2√135) / √12. First, let's simplify the radicals in the numerator. √48 = √(16 * 3) = 4√3. √135 = √(9 * 15) = 3√15. So, 2√135 = 2 * 3√15 = 6√15. Our expression becomes (4√3 - 6√15) / √12. Now, √12 = √(4 * 3) = 2√3. Thus, we have (4√3 - 6√15) / (2√3). Dividing each term by 2√3, we get (4√3 / 2√3) - (6√15 / 2√3) = 2 - 3√(15/3) = 2 - 3√5. The simplified expression is 2 - 3√5. Again, dividing radicals and then simplifying helps in reaching the solution systematically.

e) -√24 (-8√3 + √150 - 3√200)

Distribute -√24 across the terms inside the parenthesis: -√24 * (-8√3) - √24 * √150 + √24 * (3√200). Now let’s simplify each term. -√24 * (-8√3) = 8√(24 * 3) = 8√72 = 8√(36 * 2) = 8 * 6√2 = 48√2. -√24 * √150 = -√(24 * 150) = -√3600 = -60. √24 * (3√200) = 3√(24 * 200) = 3√4800 = 3√(1600 * 3) = 3 * 40√3 = 120√3. Putting it all together, we have 48√2 - 60 + 120√3. So, the simplified expression is 48√2 - 60 + 120√3. Distributing carefully, and then simplifying the radicals step by step, is key.

f) 2√1782 : (√792 - √550)

This can be written as 2√1782 / (√792 - √550). First, let's simplify each radical. √1782 = √(81 * 22) = 9√22. √792 = √(36 * 22) = 6√22. √550 = √(25 * 22) = 5√22. So, we have 2 * 9√22 / (6√22 - 5√22) = 18√22 / (√22) = 18. Therefore, the simplified expression is 18. Factoring out the common radical before doing the division simplifies the entire calculation. This shows a clever trick of identifying common factors within the radicals to simplify calculation.

Alright, there you have it! We've tackled each of those radical expressions and simplified them down to their core. Remember, practice is essential when it comes to mastering radical simplification. Keep working on these types of problems, and you'll become much more confident in your ability to manipulate and simplify radicals. And that's a wrap! Keep practicing and have fun with math! You've got this!