Finding Integer Values Of X Where √(48/x) Is A Whole Number

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Hey guys! Today, we're diving into a fun math problem that involves square roots, fractions, and positive integers. The core of our challenge? Figuring out how many different values of x will make the expression √(48/x) a whole number, given that x itself is a positive integer. Sounds like a puzzle, right? Let's break it down and solve it together!

Understanding the Problem

So, what does it really mean for √(48/x) to be a whole number? Well, first off, a whole number is a non-negative integer (0, 1, 2, 3, and so on). For the square root of a fraction to be a whole number, the fraction itself must be a perfect square (like 1, 4, 9, 16, etc.). This is our guiding principle. Therefore, our main keyword to focus on is determining the conditions under which 48/x results in a perfect square. We need to identify the values of x that allow 48 to be divided in such a way that the quotient is a perfect square. This involves understanding the factors of 48 and how they can be manipulated to form perfect squares. Let's dive a little deeper into how we can achieve this.

To kick things off, we need to consider the prime factorization of 48. This will help us break down the number into its fundamental components and see how different values of x can affect the outcome. Remember, prime factorization is expressing a number as a product of its prime factors. For 48, this is 2 × 2 × 2 × 2 × 3, or 2⁴ × 3. Now, here's where it gets interesting. We want 48/x to be a perfect square. Perfect squares have a neat property: all the exponents in their prime factorization are even. Think about it: 4 (which is 2²) has an exponent of 2, 9 (which is 3²) has an exponent of 2, and so on. So, when we divide 48 by x, we need to make sure that the resulting exponents are all even. This gives us a crucial insight into how we should approach finding the values of x. It's like we're setting up a mathematical puzzle where the pieces (factors of 48) need to fit together perfectly to create a perfect square.

Finding the Factors of 48

The next step in our quest is to list out all the factors of 48. Why? Because x must be one of these factors if it's going to divide 48 evenly. Listing the factors helps us systematically explore the possible values of x and see which ones make √(48/x) a whole number. Factors come in pairs, which makes our job a little easier. Let’s start by identifying these pairs. We know that 1 and 48 are factors, as are 2 and 24. We continue this process to find all the numbers that divide 48 without leaving a remainder. By identifying all the factors, we lay the groundwork for determining which values of x will lead to a perfect square under the square root. This step is crucial because it narrows down the possibilities and allows us to focus on the factors that truly matter for our problem.

The factors of 48 are: 1, 2, 3, 4, 6, 8, 12, 16, 24, and 48. Now we have a concrete list to work with! We can go through each of these factors and check whether dividing 48 by them results in a number that is a perfect square. This is where the fun really begins, as we start to see how the different factors interact with the goal of creating a perfect square. By examining each factor, we’re essentially testing different scenarios and seeing which ones fit the criteria we established earlier. This process of elimination and verification is a core part of problem-solving in mathematics, and it helps us develop a deeper understanding of the relationships between numbers.

Determining Values of x

Now comes the exciting part: testing each factor to see if 48 divided by that factor results in a perfect square. This is where we put our detective hats on and systematically go through our list. We'll take each factor of 48 and divide 48 by it. If the result is a perfect square, then that factor is a valid value for x. Remember, a perfect square is a number that can be obtained by squaring an integer (e.g., 1, 4, 9, 16). So, we're looking for quotients that fit this description. This step is a practical application of our earlier understanding of perfect squares and factors, and it’s where we start to see the solution take shape. It's like piecing together a puzzle, where each factor is a potential piece that may or may not fit.

Let's go through the factors one by one:

  • If x = 1, then 48/x = 48/1 = 48. Is 48 a perfect square? Nope.
  • If x = 2, then 48/x = 48/2 = 24. Is 24 a perfect square? Nope.
  • If x = 3, then 48/x = 48/3 = 16. Is 16 a perfect square? Yes! (4² = 16)
  • If x = 4, then 48/x = 48/4 = 12. Is 12 a perfect square? Nope.
  • If x = 6, then 48/x = 48/6 = 8. Is 8 a perfect square? Nope.
  • If x = 8, then 48/x = 48/8 = 6. Is 6 a perfect square? Nope.
  • If x = 12, then 48/x = 48/12 = 4. Is 4 a perfect square? Yes! (2² = 4)
  • If x = 16, then 48/x = 48/16 = 3. Is 3 a perfect square? Nope.
  • If x = 24, then 48/x = 48/24 = 2. Is 2 a perfect square? Nope.
  • If x = 48, then 48/x = 48/48 = 1. Is 1 a perfect square? Yes! (1² = 1)

From this, we can see that there are three values of x that make 48/x a perfect square. This systematic approach not only gives us the answer but also reinforces our understanding of the concepts involved. It’s a great example of how breaking down a problem into smaller steps can make it much more manageable.

The Solution

So, after our thorough investigation, we've found that there are three values of x that make √(48/x) a whole number. These values are 3, 12, and 48. Remember, we arrived at this solution by first understanding what it means for a square root to be a whole number, then identifying the factors of 48, and finally testing each factor to see if it resulted in a perfect square when dividing 48. This step-by-step approach is a powerful tool for tackling math problems, and it's something you can apply to all sorts of challenges.

Therefore, the answer to our original question is that there are 3 different values of x for which √(48/x) is a whole number. This concludes our journey through this particular math problem. We not only found the solution but also reinforced our understanding of factors, perfect squares, and problem-solving strategies. Math is not just about finding answers; it's about the process of discovery and the connections we make along the way.

Conclusion

Alright, guys! We successfully navigated this problem by breaking it down into manageable steps. We started by understanding the core concept – what makes a square root a whole number? Then, we used prime factorization and the factors of 48 to narrow down our possibilities. Finally, we tested each potential value of x to see if it fit the criteria. This methodical approach is super useful for tackling all sorts of math challenges. Remember, the key is to understand the problem, break it down, and work through it step by step. Keep practicing, and you'll become a math whiz in no time!