Solving The Mathematical Expression 54 + 216√(6 + √24)

Hey guys! Let's dive into solving this intriguing mathematical expression: 54 + 216√(6 + √24). Math problems can seem daunting at first, but breaking them down step by step makes them much more manageable. Our main keywords here are mathematical expressions, square roots, and simplification. So, grab your thinking caps, and let's get started!

Understanding the Expression

At its core, the expression involves addition, multiplication, and nested square roots. Specifically, we are dealing with the expression 54 + 216√(6 + √24). To tackle this effectively, we need to understand the order of operations (PEMDAS/BODMAS) and how to simplify square roots. Remember, the order of operations tells us to handle parentheses (or brackets), exponents, multiplication and division (from left to right), and then addition and subtraction (from left to right).

Breaking Down the Components

1. Constants: We have the constants 54 and 216. These are straightforward numbers that we'll use in our calculations.
2. Square Root within a Square Root: The expression includes √24 within the larger square root √(6 + √24). This is where things get a bit interesting. We need to simplify √24 first.
3. Addition inside the Square Root: The sum 6 + √24 is under the main square root, so we need to address this after simplifying √24.
4. Multiplication: The simplified square root result will be multiplied by 216.
5. Final Addition: Finally, we add the result of the multiplication to 54 to get our final answer.

Initial Steps: Simplifying √24

To simplify √24, we need to find the prime factorization of 24. The prime factorization of 24 is 2 × 2 × 2 × 3, which can be written as 2³ × 3. We can rewrite √24 as √(2² × 2 × 3). Since √a² = a, we can pull out the 2² from under the square root:

√24 = √(2² × 2 × 3) = 2√(2 × 3) = 2√6

So, √24 simplifies to 2√6. This is a crucial step in making our original expression easier to handle. Simplifying square roots like this is a fundamental skill in algebra, and it's super handy for problems like this one.

Substituting and Simplifying Further

Now that we've simplified √24 to 2√6, let's substitute it back into our original expression. Our expression now looks like this:

54 + 216√(6 + 2√6)

The next step is to simplify the expression inside the main square root, which is 6 + 2√6. This part requires a bit of algebraic intuition. We're looking for a way to rewrite this expression as a perfect square so we can eliminate the square root.

Recognizing a Perfect Square

Notice that 6 + 2√6 looks suspiciously like it could be part of a perfect square of the form (a + b)². Expanding (a + b)² gives us a² + 2ab + b². We want to match this form with 6 + 2√6. Let's break it down:

• We have a term with a square root (2√6), which corresponds to 2ab.
• We have a constant term (6), which corresponds to a² + b².

If we let a = √3 and b = √2, then:

• 2ab = 2(√3)(√2) = 2√6 (This matches our term!)
• a² + b² = (√3)² + (√2)² = 3 + 2 = 5 (Almost there! We need this to be 6).

Ah, but we made a slight mistake! What if we consider a more direct approach? We need to rewrite 6 + 2√6 in the form of (a + b)². Notice that if we think of √6 as √3 * √2, we might be onto something. Let's try to express 6 + 2√6 as (${\sqrt{x} + \sqrt{y}}$)² = x + y + 2√(xy).

Here, we need x + y = 6 and xy = 6. A-ha! Let's try x = 3 and y = 3. Nope, that doesn't quite work. But what if we manipulate it a bit differently?

The Correct Perfect Square

We need to find two numbers that add up to 6 and whose product is also related to 6. Think about perfect squares that involve square roots. If we rewrite 6 + 2√6 as (√a + √b)², it expands to a + b + 2√(ab). So, we need a + b = 6 and ab = 6. This might seem tricky, but let's try something different.

Let’s think about (√3 + √3)². That's not it because √3 + √3 is not going to give us the format we need. What if we consider (√x + √y)² = x + y + 2√(xy).

We want x + y = 6 and xy to equal 6 when rooted and multiplied by 2. It’s like we’re chasing our tail here! The key is to recognize that our initial breakdown was almost correct.

The correct perfect square lies in recognizing that we can rewrite the expression inside the square root as:

6 + 2√6 = (√3 + √2)² = (√3)² + 2(√3)(√2) + (√2)² = 3 + 2√6 + 2 = 5 + 2√6

Oops! Almost. We need that constant to be 6, not 5. Let's backtrack and rethink our approach.

Back to Basics: Reassessing the Square Root

Sometimes, the simplest approach is the best. Let's revisit the expression inside the square root: 6 + 2√6.

We need to express this in the form (a + b)², which expands to a² + 2ab + b². If we let a = √x and b = √y, then (√x + √y)² = x + y + 2√(xy).

We need x + y = 6 and √(xy) = √6, which means xy = 6.

Okay, let's think of factors of 6. We have 1 and 6, 2 and 3. If we try x = 2 and y = 3, we get:

• x + y = 2 + 3 = 5 (Not 6, so this doesn't work).

It seems we are missing a simple trick. We need to find a cleaner way to express the inside of this square root. Let’s pause and rethink our perfect square strategy.

The Eureka Moment!

Alright, guys, after some deep thought, let's hit the rewind button and reassess. Sometimes, in math, you have to take a step back to see the path forward. We have 6 + 2√6 inside the square root. Our goal is to rewrite it as something squared so that the square root simplifies nicely.

The key insight here is realizing that 6 + 2√6 doesn’t directly translate into a perfect square in an obvious way. We've been trying to force it into a mold that doesn’t quite fit. Let’s think about this differently. Is there another way to look at it?

What if we made a small algebraic manipulation? Let's multiply the entire term inside the outer square root by a clever form of 1. This might seem weird, but trust me, it’s a trick worth knowing. What if we tried to rewrite the expression to unveil a hidden structure?

After a moment of clarity, we realize that we might be overcomplicating things. The expression 6 + 2√6 is indeed tricky, but maybe, just maybe, there’s a small error in the original problem statement! Let’s proceed assuming the problem is correctly stated and look for other approaches.

Taking a Different Tack

Given our struggles to simplify 6 + 2√6 into a perfect square, let's consider a different approach. Sometimes in complex math problems, recognizing what not to do is as important as knowing what to do.

We've tried the perfect square approach extensively, and it's not yielding a straightforward result. So, let's hold off on that for a moment and see if we can simplify the entire expression in a different way.

We have 54 + 216√(6 + 2√6). Let's focus on the term 216√(6 + 2√6). If we could somehow estimate or bound the value of √(6 + 2√6), it might give us some insight. The goal here is to understand the magnitude of this term.

Bounding the Square Root

We know that √4 = 2 and √9 = 3. Since 6 is between 4 and 9, √6 is between 2 and 3. More precisely, √6 is approximately 2.45. So, 2√6 is approximately 2 * 2.45 = 4.9. Thus, 6 + 2√6 is approximately 6 + 4.9 = 10.9.

Therefore, √(6 + 2√6) is approximately √10.9, which is a bit more than √9 = 3. Let's say it's around 3.3. So, 216√(6 + 2√6) is approximately 216 * 3.3, which is around 712.8. Adding 54 to this gives us approximately 766.8.

This estimation doesn't give us an exact answer, but it provides a ballpark figure. If we were taking a multiple-choice test, this might help us narrow down the options. However, for an exact solution, we need a more precise method.

Seeking a Numerical Solution

Since we've hit a roadblock with algebraic simplification, let's explore a numerical approach. Using a calculator or computational tool to evaluate √(6 + 2√6) will give us a more precise value.

√(6 + 2√6) ≈ √(6 + 2 * 2.449) ≈ √(6 + 4.898) ≈ √10.898 ≈ 3.301

Now, we multiply this by 216: 216 * 3.301 ≈ 712.99

Finally, we add 54: 54 + 712.99 ≈ 766.99

So, the numerical solution is approximately 766.99. This suggests that the exact answer might be a whole number or a simple fraction close to this value.

Final Thoughts and Conclusion

Alright, folks! After a winding journey through square roots, perfect squares, and algebraic manipulations, we've arrived at a numerical approximation of the expression 54 + 216√(6 + √24). We've learned that sometimes, the direct algebraic approach might not be immediately clear, and it's okay to explore other strategies, like numerical estimation.

While we didn't find a perfect algebraic simplification, we gained valuable insights into the behavior of the expression. And that, my friends, is what math is all about – the journey of exploration and discovery! Our main takeaway is to keep trying different methods and not be afraid to think outside the box.

In conclusion, the result of the mathematical operation 54 + 216√(6 + √24) is approximately 766.99. If this were a test question, we’d look for an answer choice closest to this value, or perhaps the problem statement has a slight variation we might have missed. Keep those calculators handy, guys, and happy math-ing!