Integer Values Of Square Root Of 48 Divided By X

Hey there, math enthusiasts! Ever stumbled upon a problem that seems simple on the surface but hides a treasure trove of mathematical concepts? Today, we're going to dissect one such problem. We'll explore the question: "If $x$ is a positive integer, for how many different values of $x$ is $\sqrt{\frac{48}{x}}$ a whole number?" This isn't just about finding the answer; it's about the journey, the logical deductions, and the elegant dance of numbers that makes mathematics so captivating. So, buckle up, and let's dive into the world of radicals, integers, and the beauty of mathematical problem-solving!

1. Understanding the Core Question

At its heart, the question asks us to find positive integer values of $x$ that make the expression $\sqrt{\frac{48}{x}}$ a whole number. To truly grasp this, we need to break it down. What does it mean for a square root to be a whole number? It means the value inside the square root, $\frac{48}{x}$, must be a perfect square. A perfect square is an integer that can be obtained by squaring another integer (e.g., 1, 4, 9, 16, and so on). Our quest, therefore, is to find the values of $x$ that transform $\frac{48}{x}$ into one of these perfect squares. This understanding forms the bedrock of our solution. We're not just plugging in numbers randomly; we're strategically seeking values that satisfy a specific mathematical property. The question subtly weaves together the concepts of division, square roots, and perfect squares, challenging us to see the connections between them. This is the essence of mathematical thinking – recognizing patterns and relationships. The more comfortable we become with these fundamental concepts, the more adept we become at tackling complex problems. This initial step of understanding the core question is often the most crucial. It sets the direction for our solution and prevents us from getting lost in the maze of possibilities. So, let's keep this understanding in mind as we move forward, unraveling the layers of this intriguing problem.

2. Prime Factorization The Key to Unlocking Perfect Squares

Now that we know we're looking for values of $x$ that make $\frac{48}{x}$ a perfect square, the next logical step is to prime factorize 48. Why prime factorization? Because it allows us to see the building blocks of 48 and how they can be arranged to form perfect squares. The prime factorization of 48 is $2^4 \cdot 3^1$. This tells us that 48 is composed of four 2s and one 3. Remember, a perfect square has an even number of each prime factor. Think about it: the square root of a number with an even exponent will be an integer. For example, $\sqrt{2^2} = 2$, $\sqrt{2^4} = 2^2 = 4$, and so on. So, to make $\frac{48}{x}$ a perfect square, we need to ensure that when we divide 48 by $x$, the resulting quotient has even exponents in its prime factorization. This is where the magic happens! We can now strategically choose values of $x$ that "cancel out" the odd exponents in the prime factorization of 48. For instance, if $x$ has a factor of 3, it will cancel out the $3^1$ in the prime factorization of 48, potentially leaving us with a perfect square. This prime factorization technique is a powerful tool in number theory. It allows us to break down complex numbers into their fundamental components, making it easier to analyze their properties and relationships. By understanding the prime factorization of 48, we've transformed our problem from a trial-and-error approach to a systematic search for solutions. We're no longer guessing; we're strategically crafting values of $x$ that meet our criteria. This is the beauty of mathematical problem-solving – using the right tools and techniques to unlock the underlying structure of a problem.

3. Finding the Valid Values of $x$ The Systematic Approach

With the prime factorization of 48 in hand ($2^4 \cdot 3^1$), we can now systematically find the values of $x$ that make $\frac{48}{x}$ a perfect square. Remember, $x$ must be a positive integer. To make the quotient a perfect square, $x$ must "remove" the factors that prevent $\frac{48}{x}$ from having even exponents in its prime factorization. Let's analyze the possibilities:

• Case 1: $\frac{48}{x}$ is a perfect square with no factor of 3. This means $x$ must have a factor of 3 to cancel out the $3^1$ in the prime factorization of 48. So, $x$ can be $3 \cdot 2^0 = 3$, $3 \cdot 2^2 = 12$, or $3 \cdot 2^4 = 48$. These values of $x$ will result in $\frac{48}{x}$ being $2^4 = 16$, $2^2 = 4$, and $2^0 = 1$, respectively, all perfect squares.

• Case 2: $\frac{48}{x}$ is a perfect square with a factor of 3. In this case, x shouldn't have a factor of 3. Thus, $x$ can be $2^0 = 1$, $2^2 = 4$, or $2^4 = 16$. These values of $x$ will result in $\frac{48}{x}$ being $2^4 * 3 = 48$, $2^2 * 3 = 12$, and $2^0 * 3 = 3$, respectively.

We need to consider the possible combinations of powers of 2 and 3 that can divide 48. Since the exponent of 2 in the prime factorization of 48 is 4, the possible exponents of 2 in $x$ are 0, 2, and 4 (to leave even exponents for the quotient). The exponent of 3 in the prime factorization of 48 is 1, so the possible exponents of 3 in $x$ are 0 and 1.

Combining these possibilities, we get the following values for $x$:

• $x = 2^0 \cdot 3^0 = 1$
• $x = 2^2 \cdot 3^0 = 4$
• $x = 2^4 \cdot 3^0 = 16$
• $x = 2^0 \cdot 3^1 = 3$
• $x = 2^2 \cdot 3^1 = 12$
• $x = 2^4 \cdot 3^1 = 48$

This systematic approach ensures that we don't miss any possible values of $x$. By carefully considering the prime factorization and the requirements for perfect squares, we've turned a potentially daunting task into a manageable process. It's like solving a puzzle, where each step logically leads to the next. This meticulous method is a hallmark of mathematical problem-solving. It's not about rushing to the answer; it's about taking the time to understand the underlying principles and applying them strategically.

4. Counting the Solutions and Verifying the Results

Now that we've identified the possible values of $x$, the next step is to count them and, importantly, verify that they indeed satisfy the original condition. We found the following values for $x$: 1, 3, 4, 12, 16, and 48. That's a total of six different values. But let's not stop there! It's crucial to verify our results to ensure accuracy. We can do this by plugging each value of $x$ back into the original expression, $\sqrt{\frac{48}{x}}$, and checking if we get a whole number:

• For $x = 1$, $\sqrt{\frac{48}{1}} = \sqrt{48}$, which is not a whole number. (Mistake Identified!)
• For $x = 3$, $\sqrt{\frac{48}{3}} = \sqrt{16} = 4$, a whole number.
• For $x = 4$, $\sqrt{\frac{48}{4}} = \sqrt{12}$, which is not a whole number. (Mistake Identified!)
• For $x = 12$, $\sqrt{\frac{48}{12}} = \sqrt{4} = 2$, a whole number.
• For $x = 16$, $\sqrt{\frac{48}{16}} = \sqrt{3}$, which is not a whole number. (Mistake Identified!)
• For $x = 48$, $\sqrt{\frac{48}{48}} = \sqrt{1} = 1$, a whole number.

Woah! It looks like our initial analysis had some errors (which is a great reminder that verification is essential!). Let's re-evaluate our cases in Section 3 based on this verification.

Going back to our cases:

• Case 1 (Corrected): $\frac{48}{x}$ is a perfect square with no factor of 3. This is correct, we need x to