Ordering Numbers: A Math Problem With Radicals

Hey guys! Today, we're diving into a super interesting math problem that involves ordering numbers with radicals. It might look intimidating at first, but don't worry, we'll break it down step by step. The problem asks us to order the numbers a, b, c, and d in descending order, where each number is defined by a complex expression involving square roots. So, buckle up, and let's get started!

Understanding the Problem

Before we jump into solving, let's make sure we fully understand what we're dealing with. We have four numbers, a, b, c, and d, each represented by a formula that includes square roots of various numbers. Our mission is to simplify these expressions and then arrange the resulting values from largest to smallest. This involves a good grasp of radical simplification and arithmetic operations. The key here is to remember the properties of square roots: √(a*b) = √a * √b and √a² = a. We also need to be comfortable with basic arithmetic operations such as addition, subtraction, multiplication, and division. Pay close attention to the order of operations (PEMDAS/BODMAS) to ensure accurate calculations. This is a classic example of a problem that tests not just your algebraic skills but also your ability to carefully apply mathematical rules. Missteps in simplifying radicals or in performing arithmetic can easily lead to incorrect answers. So, let's take our time and approach this systematically, ensuring each step is precise and well-understood. This kind of problem often appears in algebra and pre-calculus courses, so mastering the techniques involved will be highly beneficial for your mathematical journey. Remember, mathematics is like building a tower; each step relies on the previous one, and a solid foundation is crucial for success. Let's start building that foundation now!

Solving for a

Let's start with a. The expression for a is: a = √24 (3√50 + 6√48 - 2√98 - 4√108). Our first step is to simplify the radicals inside the parentheses. We can rewrite each radical by factoring out perfect squares. For example, √50 can be written as √(25 * 2) which simplifies to 5√2. Similarly, √48 becomes √(16 * 3) or 4√3, √98 becomes √(49 * 2) or 7√2, and √108 becomes √(36 * 3) or 6√3. Substituting these simplified radicals back into the expression, we get: a = √24 (3 * 5√2 + 6 * 4√3 - 2 * 7√2 - 4 * 6√3). Now, we simplify further by performing the multiplications inside the parentheses: a = √24 (15√2 + 24√3 - 14√2 - 24√3). Notice that we have like terms (terms with the same radical) that can be combined. The 24√3 and -24√3 terms cancel each other out, and we can combine 15√2 and -14√2 to get √2. So, the expression simplifies to: a = √24 (√2). Finally, we can rewrite √24 as √(4 * 6) or 2√6. Substituting this back into the expression gives us: a = 2√6 * √2 = 2√(6 * 2) = 2√12. We can simplify √12 further as √(4 * 3) or 2√3. Thus, a = 2 * 2√3 = 4√3. So, after all those simplifications, we find that a = 4√3. That wasn't so bad, right? Let's keep the momentum going and tackle the next expression.

Solving for b

Next up, we have b, which is given by the expression: b = -2√3 (2√32 - 4√72 - √800 + 3√162) / 6. Just like with a, our first task is to simplify the radicals inside the parentheses. Let's break it down: √32 can be written as √(16 * 2) or 4√2, √72 becomes √(36 * 2) or 6√2, √800 becomes √(400 * 2) or 20√2, and √162 becomes √(81 * 2) or 9√2. Now, let's substitute these simplified radicals back into the expression: b = -2√3 (2 * 4√2 - 4 * 6√2 - 20√2 + 3 * 9√2) / 6. Next, we perform the multiplications inside the parentheses: b = -2√3 (8√2 - 24√2 - 20√2 + 27√2) / 6. Now, let's combine the like terms inside the parentheses. We have several terms with √2, so we can add their coefficients: 8 - 24 - 20 + 27 = -9. So the expression inside the parentheses simplifies to -9√2. Substituting this back into the equation for b, we get: b = -2√3 (-9√2) / 6. Now, let's multiply the terms in the numerator: -2√3 * -9√2 = 18√(3 * 2) = 18√6. So, b = 18√6 / 6. Finally, we can simplify this fraction by dividing 18 by 6, which gives us 3. Thus, b = 3√6. Great job! We've successfully simplified b. Now, let's move on to the next number and keep this simplification train rolling!

Solving for c

Now, let's tackle c. The expression for c is a bit more complex: c = [√216 (2√200 - 3√512 + √486 + 5√72) - 324] / (3√3). As before, the first step is to simplify the radicals inside the parentheses and elsewhere. Let's start with the radicals inside the main parentheses: √200 can be written as √(100 * 2) or 10√2, √512 becomes √(256 * 2) or 16√2, √486 becomes √(81 * 6) or 9√6, and √72 becomes √(36 * 2) or 6√2. Also, we need to simplify √216, which can be written as √(36 * 6) or 6√6. Now, let's substitute these simplified radicals back into the expression: c = [6√6 (2 * 10√2 - 3 * 16√2 + 9√6 + 5 * 6√2) - 324] / (3√3). Next, perform the multiplications inside the parentheses: c = [6√6 (20√2 - 48√2 + 9√6 + 30√2) - 324] / (3√3). Combine the like terms (the √2 terms) inside the parentheses: 20 - 48 + 30 = 2. So the expression inside the parentheses simplifies to 2√2 + 9√6. Now, substitute this back into the equation for c: c = [6√6 (2√2 + 9√6) - 324] / (3√3). Distribute the 6√6 across the terms inside the parentheses: 6√6 * 2√2 = 12√(6 * 2) = 12√12 = 12 * 2√3 = 24√3, and 6√6 * 9√6 = 54 * 6 = 324. So, the expression becomes: c = [24√3 + 324 - 324] / (3√3). Notice that the +324 and -324 terms cancel each other out, leaving us with: c = 24√3 / (3√3). Finally, we can simplify this fraction by dividing 24 by 3, which gives us 8. The √3 terms also cancel out. Thus, c = 8. Awesome! We've conquered c. Only one more to go! Let's keep up the great work and simplify the last expression.

Solving for d

Alright, let's tackle the last one, d. The expression for d is: d = 3√8 (2√150 - 3√384 + 5√54 - 3√24 + 2√96) / 6. You know the drill by now – let's start by simplifying those radicals! √8 can be written as √(4 * 2) or 2√2, √150 becomes √(25 * 6) or 5√6, √384 becomes √(64 * 6) or 8√6, √54 becomes √(9 * 6) or 3√6, √24 becomes √(4 * 6) or 2√6, and √96 becomes √(16 * 6) or 4√6. Let's substitute these simplified radicals back into the expression: d = 3 * 2√2 (2 * 5√6 - 3 * 8√6 + 5 * 3√6 - 3 * 2√6 + 2 * 4√6) / 6. Simplify the expression: d = 6√2 (10√6 - 24√6 + 15√6 - 6√6 + 8√6) / 6. Combine the like terms (the √6 terms) inside the parentheses: 10 - 24 + 15 - 6 + 8 = 3. So the expression inside the parentheses simplifies to 3√6. Now, substitute this back into the equation for d: d = 6√2 (3√6) / 6. Multiply the terms in the numerator: 6√2 * 3√6 = 18√(2 * 6) = 18√12 = 18 * 2√3 = 36√3. So, d = 36√3 / 6. Finally, simplify this fraction by dividing 36 by 6, which gives us 6. Thus, d = 6√3. Woohoo! We've successfully simplified all four expressions. You're doing fantastic!

Ordering the Numbers

Okay, now that we've simplified a, b, c, and d, let's put them in descending order. We have:

• a = 4√3
• b = 3√6
• c = 8
• d = 6√3

To easily compare these numbers, it's helpful to either convert them to decimal approximations or to compare their squares. Let's compare their squares. Squaring each number, we get:

• a² = (4√3)² = 16 * 3 = 48
• b² = (3√6)² = 9 * 6 = 54
• c² = 8² = 64
• d² = (6√3)² = 36 * 3 = 108

From the squares, we can easily see the order: 108 > 64 > 54 > 48. Therefore, in descending order, the numbers are:

d > c > b > a

So, there you have it! We've successfully ordered the numbers a, b, c, and d in descending order. You nailed it!

Final Thoughts

This problem was a fantastic exercise in simplifying radicals and comparing numbers. You've tackled complex expressions, applied the rules of radicals, and successfully ordered the results. Give yourself a pat on the back – you've earned it! Remember, practice makes perfect, so keep honing your skills, and you'll become a math whiz in no time. Keep up the awesome work, and I'll see you in the next math adventure!