Solve L=[√(3⁴√(27.∛81))]^(24/25): Step-by-Step Solution

by Kenji Nakamura 56 views

Okay, guys, let's dive into this mathematical expression and figure out how to solve it! We're dealing with some nested radicals and exponents here, but don't worry, we'll take it step by step. The problem is to simplify: L=[√(3⁴√(27.∛81))]^(24/25). Our goal is to find the value of L and choose the correct answer from the options provided: a) 3, b) 5, c) 2, d) 9, e) 27. To solve this problem effectively, we'll need to use our knowledge of radicals, exponents, and how they interact with each other. It might look intimidating at first glance, but with a systematic approach, we can definitely crack this! Remember, math problems like these are like puzzles, and each step we take is a piece falling into place. So, let's roll up our sleeves and get started! We'll begin by simplifying the innermost parts of the expression and work our way outwards. This is a classic strategy for dealing with complex expressions, and it will help us keep things organized and avoid mistakes. Are you ready to see how it's done? Let's go!

Step-by-Step Solution

First, let's focus on the innermost part: ∛81. We need to find the cube root of 81. To do this, it helps to express 81 as a power of 3. We know that 81 is 3 to the power of 4 (3⁴ = 81). So, ∛81 is the same as ∛(3⁴). Now, remember that a cube root is the same as raising to the power of 1/3. Therefore, ∛(3⁴) can be written as (3⁴)^(1/3). Using the rule of exponents that says (am)n = a^(m*n), we can simplify this to 3^(4/3). So, the cube root of 81 is 3 raised to the power of 4/3. We've made good progress already! Next, we need to multiply this result by 27. Remember, 27 is 3 cubed (3³). So, we're now dealing with 27 * ∛81, which we've just figured out is 3³ * 3^(4/3). When multiplying numbers with the same base, we add their exponents. So, 3³ * 3^(4/3) becomes 3^(3 + 4/3). To add these exponents, we need a common denominator. 3 can be written as 9/3, so we have 3^(9/3 + 4/3), which simplifies to 3^(13/3). Great! We've simplified the expression inside the fourth root.

Now, let's tackle that fourth root: ⁴√(27.∛81). We've already determined that 27.∛81 is 3^(13/3). So, we're looking for the fourth root of 3^(13/3), which can be written as ⁴√(3^(13/3)). Just like with the cube root, taking the fourth root is the same as raising to the power of 1/4. So, we have (3(13/3))(1/4). Again, using the rule of exponents (am)n = a^(m*n), we multiply the exponents: (13/3) * (1/4) = 13/12. Therefore, ⁴√(27.∛81) simplifies to 3^(13/12). We're getting closer to the final answer! The next step is to multiply this by the square root. So, we need to consider √(3 * ⁴√(27.∛81)). We know that ⁴√(27.∛81) is 3^(13/12), so we have √(3 * 3^(13/12)). Remember, 3 is the same as 3¹, so we're looking at √(3¹ * 3^(13/12)). When multiplying with the same base, we add the exponents: 3^(1 + 13/12). 1 can be written as 12/12, so we have 3^(12/12 + 13/12), which equals 3^(25/12). But don't forget, we still need to take the square root! Taking the square root is the same as raising to the power of 1/2. So, √(3^(25/12)) is (3(25/12))(1/2). Multiplying the exponents, we get 3^((25/12) * (1/2)) = 3^(25/24). We've now simplified the entire expression inside the brackets!

Finally, we need to raise this entire result to the power of 24/25: L = [3(25/24)](24/25). This looks much simpler now, doesn't it? Once again, we use the rule of exponents (am)n = a^(m*n). We multiply the exponents: (25/24) * (24/25). Notice something cool here? The 25 in the numerator and denominator cancel out, and the 24 in the numerator and denominator also cancel out! This leaves us with 1. So, L = 3¹ which simply equals 3. We've done it! The value of L is 3. Looking back at our options, the correct answer is a) 3.

Key Concepts Used

Throughout this problem, we've used several key mathematical concepts. Let's recap them to solidify our understanding:

  • Radicals and Exponents: We've seen how radicals (like square roots, cube roots, and fourth roots) can be expressed as fractional exponents. This is a crucial connection to understand for simplifying expressions like this. For example, √a = a^(1/2), ∛a = a^(1/3), and so on.
  • Rules of Exponents: These rules are the bread and butter of simplifying exponential expressions. We've heavily used two main rules:
    • (am)n = a^(m*n): This rule allows us to simplify expressions where a power is raised to another power.
    • a^m * a^n = a^(m+n): This rule helps us when multiplying numbers with the same base. We add the exponents.
  • Fraction Operations: Dealing with fractional exponents requires a good grasp of fraction addition and multiplication. We needed to find common denominators to add fractions and multiply fractions correctly.

By mastering these concepts, you'll be well-equipped to tackle similar problems involving radicals and exponents. Remember, practice makes perfect! The more you work with these concepts, the more comfortable and confident you'll become.

Common Mistakes to Avoid

When dealing with problems like this, it's easy to make small errors that can throw off your entire solution. Let's discuss some common pitfalls to watch out for:

  • Incorrectly Applying Exponent Rules: The rules of exponents are powerful, but they need to be applied carefully. A common mistake is to mix up the rules for multiplying exponents versus adding them. Remember, (am)n = a^(m*n) (multiply exponents) and a^m * a^n = a^(m+n) (add exponents).
  • Forgetting the Order of Operations: Just like with any mathematical expression, the order of operations (PEMDAS/BODMAS) is crucial. Make sure you're simplifying the innermost expressions first and working your way outwards. Ignoring the order of operations can lead to incorrect simplifications.
  • Errors with Fractions: Fractional exponents can be tricky. Double-check your fraction arithmetic, especially when adding or multiplying fractions. A small mistake with fractions can propagate through the entire problem.
  • Rushing the Process: These problems often involve multiple steps. It's tempting to rush through them, but taking your time and being methodical is key to accuracy. Write out each step clearly to minimize the chance of errors.
  • Not Simplifying Enough: Sometimes, you might stop simplifying before you reach the simplest form. Always double-check if there are further simplifications possible, especially with exponents.

By being aware of these common mistakes, you can consciously avoid them and increase your chances of solving the problem correctly.

Practice Problems

To really master these concepts, it's important to practice! Here are a couple of similar problems you can try on your own:

  1. Simplify: [√(2³√(16.∛64))]^(15/16)
  2. Evaluate: [(5²√(125.⁴√25))]^(8/9)

Work through these problems step by step, using the techniques we've discussed. Check your answers carefully and don't be afraid to review the solution if you get stuck. The more you practice, the more natural these types of problems will become. Remember, math is a skill that improves with practice, just like anything else!

Conclusion

We've successfully navigated this complex expression and found that L = 3. By breaking down the problem into smaller, manageable steps and carefully applying the rules of exponents and radicals, we were able to arrive at the correct solution. Remember, the key to success with these types of problems is a methodical approach, a solid understanding of the fundamental concepts, and plenty of practice. So, keep practicing, guys, and you'll be conquering these mathematical challenges in no time! Math can be challenging, but it's also incredibly rewarding when you finally crack a tough problem. Keep that in mind as you continue your mathematical journey. You've got this!