Math Problem: Solving $4^{-1/2} imes \sqrt{32}$

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Math Problem: Solving $4^{-1/2} \times \sqrt{32}$

Hey math enthusiasts! Let's dive into a fun problem. We're gonna solve the expression: 4−12×324 {}^{ - \frac{1}{2} } \times \sqrt{32}. It might look a little intimidating at first glance, but trust me, it's totally manageable. We'll break it down step by step, making it super clear and easy to follow. Get ready to flex those math muscles!

Understanding the Basics: Exponents and Square Roots

Alright, before we jump into the main problem, let's quickly recap some key concepts. This will help us avoid any confusion. First up, exponents. The negative exponent in 4−1/24^{-1/2} means we need to take the reciprocal of the base (which is 4) and raise it to the power of 1/2. So, 4−1/24^{-1/2} is the same as 141/2\frac{1}{4^{1/2}}. Basically, a negative exponent flips the number over, putting it in the denominator. It's like the number is going on an adventure and flipping upside down!

Next, we've got square roots. The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3, because 3 * 3 = 9. In our problem, we have 32\sqrt{32}. We need to figure out what number, when multiplied by itself, gives us 32. It's a little trickier than a perfect square like 9, but don't worry, we'll get there. Knowing these concepts is super important for solving the expression, so make sure you've got them down. It’s like having the right tools before starting a project – makes everything much smoother, right?

Now, let's put these concepts to work and start solving the problem. We will start with simplifying each part of the expression.

Step-by-Step Solution: Breaking Down the Problem

Alright, let's get our hands dirty and actually solve this math problem. We'll take it one step at a time, so you can follow along easily. Here's how we'll do it:

  1. Simplify 4−1/24^{-1/2}: As we discussed earlier, 4−1/24^{-1/2} means 141/2\frac{1}{4^{1/2}}. And what's 41/24^{1/2}? It's the square root of 4, which is 2. So, 4−1/24^{-1/2} simplifies to 12\frac{1}{2}. Awesome, we've simplified the first part of our expression.
  2. Simplify 32\sqrt{32}: Now, let's tackle 32\sqrt{32}. We can simplify this by finding the largest perfect square that divides 32. That would be 16, since 16 * 2 = 32. So, we can rewrite 32\sqrt{32} as 16×2\sqrt{16 \times 2}. Using the properties of square roots, we can split this up into 16×2\sqrt{16} \times \sqrt{2}. The square root of 16 is 4, so we get 424\sqrt{2}. We've simplified the second part! Nice work.
  3. Multiply the Simplified Terms: Now that we've simplified both parts of the expression, it's time to multiply them together. We have 12×42\frac{1}{2} \times 4\sqrt{2}. Multiply the numbers together: 12×4=2\frac{1}{2} \times 4 = 2. So, our expression becomes 222\sqrt{2}. And there you have it – we've solved the problem!

As you can see, solving this problem is all about breaking it down into smaller, more manageable steps. Don’t be afraid to take your time and double-check your work. It's like building with LEGOs – take it one block at a time, and you'll eventually have something amazing!

Identifying the Correct Answer Choice

Now that we've worked out the answer, let's see which of the answer choices matches our solution, which is 222\sqrt{2}. Looking at the provided options, we need to match our calculated answer. Remember, the options could be in a different order, so careful examination is key. We are looking for the option that represents 222\sqrt{2}. Let's quickly review the choices to identify the correct one. The answer choices are:

A) 8 B) 4 C) 222\sqrt{2} D) 22\frac{\sqrt{2}}{2} E) 24\frac{\sqrt{2}}{4}

From the choices provided, option C) 222\sqrt{2} is the correct answer. This is because it is the simplified form of the original equation we solved. The other options do not match the correct answer after the equation is fully simplified. Congrats, you successfully solved the problem and found the right answer choice!

Tips for Similar Math Problems

Alright, guys, you crushed it! You successfully solved the problem 4−1/2×324^{-1/2} \times \sqrt{32}. Here are some quick tips for tackling similar math problems in the future:

  • Remember the Order of Operations: Always follow the order of operations (PEMDAS/BODMAS): Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). This ensures you solve the problem in the correct sequence.
  • Simplify Step by Step: Break down complex problems into smaller, easier-to-manage steps. This helps avoid confusion and makes the problem less daunting.
  • Practice Regularly: The more you practice, the more comfortable you'll become with these types of problems. Try similar examples and variations to reinforce your understanding.
  • Understand the Concepts: Make sure you have a solid grasp of exponents, square roots, and other fundamental math concepts. This will make problem-solving much easier.
  • Double-Check Your Work: Always double-check your calculations to avoid silly mistakes. It's easy to overlook a small detail, so taking an extra moment to review your work can save you from errors.
  • Use a Calculator Wisely: While calculators can be helpful, make sure you understand the underlying concepts. Use the calculator to check your answers, but don't rely on it to solve the entire problem for you.
  • Don't Be Afraid to Ask for Help: If you get stuck, don't hesitate to ask your teacher, a friend, or an online resource for help. It's better to clarify any confusion early on.

Keep practicing, stay curious, and you'll become a math whiz in no time. Keep the tips in mind and you'll be well-equipped to tackle similar problems with confidence. Happy solving, and keep up the great work! You've got this!

Conclusion: You've Got This!

So there you have it! We've successfully solved the math problem 4−1/2×324^{-1/2} \times \sqrt{32}, identified the correct answer choice (222\sqrt{2}), and learned some valuable tips for future problems. Remember, math can be fun and rewarding, especially when you break it down into manageable steps. Keep practicing, stay curious, and don't be afraid to challenge yourself. You've got this, and I'm super proud of your efforts!

Keep up the great work, and happy solving! Until next time, keep those math muscles flexed!