If $x$ Is A Positive Integer, For How Many Different Values Of $x$ Is $\sqrt{\frac{48}{x}}$ A Whole Number?A. 2 B. 3 C. 6 D. 10

Feb 26, 2025 by ADMIN 137 views

Solving the Mystery of Whole Numbers: A Mathematical Exploration

In the realm of mathematics, whole numbers play a vital role in various mathematical operations and equations. The concept of whole numbers is often associated with integers, which are positive or negative numbers without a fractional part. In this article, we will delve into the world of whole numbers and explore the conditions under which the expression $\sqrt{\frac{48}{x}}$ yields a whole number. We will examine the possible values of $x$ that satisfy this condition and determine the number of different values of $x$ that meet the criteria.

The problem at hand involves finding the number of different values of $x$ for which the expression $\sqrt{\frac{48}{x}}$ is a whole number. To approach this problem, we need to understand the properties of square roots and the conditions under which they yield whole numbers. A whole number is a positive integer that is not a fraction or a decimal. In other words, it is an integer that is greater than zero and has no fractional part.

The Role of Square Roots

Square roots are mathematical operations that involve finding the number that, when multiplied by itself, gives a specified value. In this case, we are dealing with the square root of the fraction $\frac{48}{x}$ . For the expression $\sqrt{\frac{48}{x}}$ to be a whole number, the value inside the square root must be a perfect square. A perfect square is a number that can be expressed as the product of an integer with itself.

Perfect Squares and Whole Numbers

To determine the number of different values of $x$ for which the expression $\sqrt{\frac{48}{x}}$ is a whole number, we need to find the perfect squares that are less than or equal to 48. The perfect squares less than or equal to 48 are 1, 4, 9, 16, 25, 36, and 49. However, we need to consider the values of $x$ that make the expression $\sqrt{\frac{48}{x}}$ a whole number.

Finding the Values of $x$

To find the values of $x$ that make the expression $\sqrt{\frac{48}{x}}$ a whole number, we need to consider the perfect squares that are factors of 48. The factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24, and 48. We need to find the perfect squares that are factors of 48 and determine the corresponding values of $x$ .

The Perfect Squares that are Factors of 48

The perfect squares that are factors of 48 are 1, 4, and 16. These perfect squares correspond to the values of $x$ that make the expression $\sqrt{\frac{48}{x}}$ a whole number.

Determining the Values of $x$

To determine the values of $x$ that correspond to the perfect squares 1, 4, and 16, we need to consider the following:

For the perfect square 1, the value of $x$ is 48.
For the perfect square 4, the value of $x$ is 12.
For the perfect square 16, the value of $x$ is 3.

In conclusion, the expression $\sqrt{\frac{48}{x}}$ is a whole number for the values of $x$ that are 48, 12, and 3. Therefore, the number of different values of $x$ for which the expression $\sqrt{\frac{48}{x}}$ is a whole number is 3.

The final answer is 3.
Frequently Asked Questions: Understanding the Mystery of Whole Numbers

In our previous article, we explored the conditions under which the expression $\sqrt{\frac{48}{x}}$ yields a whole number. We determined that the number of different values of $x$ for which the expression $\sqrt{\frac{48}{x}}$ is a whole number is 3. In this article, we will address some of the frequently asked questions related to this topic.

Q: What is the significance of the expression $\sqrt{\frac{48}{x}}$ ?

A: The expression $\sqrt{\frac{48}{x}}$ is significant because it involves the concept of square roots and perfect squares. The expression is a whole number when the value inside the square root is a perfect square.

Q: What are perfect squares?

A: Perfect squares are numbers that can be expressed as the product of an integer with itself. For example, 1, 4, 9, 16, 25, 36, and 49 are perfect squares.

Q: How do we determine the number of different values of $x$ for which the expression $\sqrt{\frac{48}{x}}$ is a whole number?

A: To determine the number of different values of $x$ for which the expression $\sqrt{\frac{48}{x}}$ is a whole number, we need to find the perfect squares that are factors of 48. The factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24, and 48. We then need to consider the perfect squares that are factors of 48 and determine the corresponding values of $x$ .

Q: What are the perfect squares that are factors of 48?

A: The perfect squares that are factors of 48 are 1, 4, and 16. These perfect squares correspond to the values of $x$ that make the expression $\sqrt{\frac{48}{x}}$ a whole number.

Q: What are the values of $x$ that correspond to the perfect squares 1, 4, and 16?

A: The values of $x$ that correspond to the perfect squares 1, 4, and 16 are 48, 12, and 3, respectively.

Q: How many different values of $x$ are there for which the expression $\sqrt{\frac{48}{x}}$ is a whole number?

A: There are 3 different values of $x$ for which the expression $\sqrt{\frac{48}{x}}$ is a whole number.

Q: What is the final answer to the problem?

A: The final answer to the problem is 3.

In conclusion, the expression $\sqrt{\frac{48}{x}}$ is a whole number for the values of $x$ that are 48, 12, and 3. We hope that this article has provided a clear understanding of the concept of whole numbers and the conditions under which the expression $\sqrt{\frac{48}{x}}$ yields a whole number.

The final answer is 3.

For further information on the topic of whole numbers and square roots, we recommend the following resources:

We hope that this article has been helpful in providing a clear understanding of the concept of whole numbers and the conditions under which the expression $\sqrt{\frac{48}{x}}$ yields a whole number.