Decoding √(A/6) A Mathematical Journey
Hey guys! Ever stumbled upon a mathematical expression that looks like it belongs in a secret code? Well, today we're diving deep into one such intriguing equation. We're going to break down the expression for A, unravel its value, and then embark on a quest to find the square root of A divided by 6. Sounds like an adventure, right? Let's get started!
Unmasking the Value of A
Our journey begins with the expression: A = √((1/2) * (1/2) * (1/3) * (1/3) * (1/6) * (3/6)). At first glance, it might seem a bit daunting, but fear not! We're going to tackle this step by step.
Cracking the Code: Multiplication Under the Radical
The first key step in deciphering this expression is to simplify the multiplication within the square root. Remember, when we're multiplying fractions, we multiply the numerators (the top numbers) together and the denominators (the bottom numbers) together. So, let's rewrite the expression under the square root as a single fraction:
(1 * 1 * 1 * 1 * 3) / (2 * 2 * 3 * 3 * 6 * 6)
Calculating the products, we get:
3 / (4 * 9 * 36) = 3 / 1296
Now our expression for A looks a little cleaner: A = √(3/1296). But we're not done yet! We can simplify this fraction further.
Simplifying the Fraction: A Quest for the Lowest Terms
Both the numerator (3) and the denominator (1296) are divisible by 3. Dividing both by 3, we get:
1 / 432
So, our expression for A now transforms into: A = √(1/432). We're making progress, guys! But the square root still looms large. It's time to confront it.
Taming the Square Root: Unveiling A's True Identity
To get rid of the square root, we need to find the square root of both the numerator and the denominator separately. The square root of 1 is easy – it's just 1. But what about the square root of 432? This is where things get interesting.
We need to find the largest perfect square that divides 432. A perfect square is a number that can be obtained by squaring an integer (e.g., 4, 9, 16, 25, etc.). Let's break down 432 into its prime factors. Prime factorization is like taking a number apart piece by piece until you are left with prime numbers (numbers only divisible by 1 and themselves).
432 = 2 * 216 = 2 * 2 * 108 = 2 * 2 * 2 * 54 = 2 * 2 * 2 * 2 * 27 = 2 * 2 * 2 * 2 * 3 * 9 = 2 * 2 * 2 * 2 * 3 * 3 * 3
So, the prime factorization of 432 is 2^4 * 3^3. We can rewrite this as (2^4 * 3^2) * 3, which is (16 * 9) * 3 or 144 * 3. Aha! 144 is a perfect square (12 * 12). This is the key we need.
Now we can rewrite √(1/432) as √(1/(144 * 3)). Using the property that √(a * b) = √a * √b, we can separate the square root: √(1/144) * √(1/3). The square root of 1/144 is 1/12. So, A = (1/12) * √(1/3).
While technically correct, it's more conventional to rationalize the denominator (get rid of the square root in the denominator). To do this, we multiply both the numerator and denominator of √(1/3) by √3. This gives us:
√(1/3) = (√1 / √3) * (√3 / √3) = √3 / 3
Therefore, A = (1/12) * (√3 / 3) = √3 / 36. We've finally cracked the code and found the simplified value of A!
The Final Quest: √ (A/6)
Now that we know A = √3 / 36, our final challenge is to find the square root of A divided by 6. Let's break it down. We need to find √ (A/6) = √ ((√3 / 36) / 6).
Dividing by 6: Simplifying the Expression
Dividing by 6 is the same as multiplying by 1/6. So, we have:
(√3 / 36) / 6 = (√3 / 36) * (1/6) = √3 / (36 * 6) = √3 / 216
Now we need to find the square root of this entire expression: √ (√3 / 216)
Taking the Square Root: Unveiling the Ultimate Answer
This looks a bit tricky, but let's think about it. We're taking the square root of a fraction where the numerator is itself a square root. To simplify this, let's rewrite the expression as:
√ (√3 / 216) = √(√3) / √216
The square root of a square root (√(√3)) is the same as taking the fourth root (∜3). So, the numerator becomes ∜3.
Now let's focus on the denominator: √216. We need to find the largest perfect square that divides 216. Let's do a prime factorization again:
216 = 2 * 108 = 2 * 2 * 54 = 2 * 2 * 2 * 27 = 2 * 2 * 2 * 3 * 9 = 2 * 2 * 2 * 3 * 3 * 3
So, the prime factorization of 216 is 2^3 * 3^3. We can rewrite this as (2^2 * 3^2) * (2 * 3), which is (4 * 9) * 6 or 36 * 6. Aha! 36 is a perfect square (6 * 6).
Therefore, √216 = √(36 * 6) = √36 * √6 = 6√6.
Putting it all together, we have:
√ (A/6) = ∜3 / (6√6)
Rationalizing the Denominator: The Final Polish
To get rid of the square root in the denominator, we multiply both the numerator and denominator by √6:
(∜3 / (6√6)) * (√6 / √6) = (∜3 * √6) / (6 * 6) = (∜3 * √6) / 36
This is the most simplified form of our answer. So, the value of √ (A/6) is (∜3 * √6) / 36.
Conclusion: A Mathematical Triumph!
Guys, we did it! We successfully navigated through the maze of square roots, fractions, and prime factorizations to find the value of √ (A/6). It might have seemed challenging at first, but by breaking down the problem into smaller, manageable steps, we were able to conquer it. Remember, math is like a puzzle – each piece fits together to reveal the solution. Keep practicing, keep exploring, and you'll be amazed at what you can achieve! If you have another math puzzle for me, let me know in the comments below! Keep up the great work, guys!
