If $y$ Is A Positive Integer, For How Many Different Values Of $y$ Is $\sqrt[3]{\frac{144}{y}}$ A Whole Number?A. 1 B. 2 C. 6 D. 15
Understanding the Problem
The problem requires us to find the number of different values of $y$ for which the expression $\sqrt[3]{\frac{144}{y}}$ results in a whole number. To begin, we need to understand the properties of cube roots and how they relate to whole numbers.
Properties of Cube Roots
A cube root of a number is a value that, when multiplied by itself twice (or cubed), gives the original number. In mathematical terms, if $x = \sqrt[3]{y}$, then $x^3 = y$. This means that the cube root of a perfect cube is a whole number.
Perfect Cubes and Whole Numbers
A perfect cube is a number that can be expressed as the cube of an integer. For example, 1, 8, 27, and 64 are perfect cubes because they can be expressed as $1^3$, $2^3$, $3^3$, and $4^3$, respectively. When we take the cube root of a perfect cube, we get a whole number.
Applying the Concept to the Problem
Now, let's apply this concept to the given expression $\sqrt[3]{\frac{144}{y}}$. We want to find the values of $y$ for which this expression results in a whole number. To do this, we need to consider the factors of 144, which is the numerator of the fraction inside the cube root.
Factors of 144
The factors of 144 are 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, and 144. These factors can be paired up to form perfect cubes, which will result in whole numbers when taken as the cube root.
Finding Perfect Cubes
To find the perfect cubes, we need to look for pairs of factors that, when multiplied together, result in a perfect cube. For example, 1 and 144 are a pair that results in a perfect cube (1 × 144 = 144 = 12^3). Similarly, 2 and 72 are a pair that results in a perfect cube (2 × 72 = 144 = 12^3).
Counting the Perfect Cubes
Let's count the number of perfect cubes that can be formed using the factors of 144. We have the following pairs:
- 1 and 144
- 2 and 72
- 3 and 48
- 4 and 36
- 6 and 24
- 8 and 18
- 9 and 16
Each of these pairs results in a perfect cube, which means that the cube root of the fraction will be a whole number. Therefore, we have 7 perfect cubes that can be formed using the factors of 144.
Conclusion
In conclusion, the number of different values of $y$ for which the expression $\sqrt[3]{\frac{144}{y}}$ results in a whole number is 15. This is because there are 15 factors of 144, and each factor can be paired up with another factor to form a perfect cube.
Final Answer
The final answer is 15.
Understanding the Problem
The problem requires us to find the number of different values of $y$ for which the expression $\sqrt[3]{\frac{144}{y}}$ results in a whole number. To begin, we need to understand the properties of cube roots and how they relate to whole numbers.
Q: What is the significance of cube roots in this problem?
A: The cube root of a number is a value that, when multiplied by itself twice (or cubed), gives the original number. In mathematical terms, if $x = \sqrt[3]{y}$, then $x^3 = y$. This means that the cube root of a perfect cube is a whole number.
Q: What are perfect cubes, and how do they relate to whole numbers?
A: A perfect cube is a number that can be expressed as the cube of an integer. For example, 1, 8, 27, and 64 are perfect cubes because they can be expressed as $1^3$, $2^3$, $3^3$, and $4^3$, respectively. When we take the cube root of a perfect cube, we get a whole number.
Q: How do we find the perfect cubes that can be formed using the factors of 144?
A: To find the perfect cubes, we need to look for pairs of factors that, when multiplied together, result in a perfect cube. For example, 1 and 144 are a pair that results in a perfect cube (1 × 144 = 144 = 12^3). Similarly, 2 and 72 are a pair that results in a perfect cube (2 × 72 = 144 = 12^3).
Q: How many perfect cubes can be formed using the factors of 144?
A: We have the following pairs:
- 1 and 144
- 2 and 72
- 3 and 48
- 4 and 36
- 6 and 24
- 8 and 18
- 9 and 16
Each of these pairs results in a perfect cube, which means that the cube root of the fraction will be a whole number. Therefore, we have 7 perfect cubes that can be formed using the factors of 144.
Q: What is the final answer to the problem?
A: The final answer is 15. This is because there are 15 factors of 144, and each factor can be paired up with another factor to form a perfect cube.
Q: Why is it important to understand the properties of cube roots and perfect cubes in this problem?
A: Understanding the properties of cube roots and perfect cubes is crucial in solving this problem because it allows us to identify the values of $y$ for which the expression $\sqrt[3]{\frac{144}{y}}$ results in a whole number.
Q: Can you provide an example of how to apply this concept to a different problem?
A: Yes, consider the expression $\sqrt[3]{\frac{27}{y}}$. To find the values of $y$ for which this expression results in a whole number, we need to consider the factors of 27, which are 1, 3, 9, and 27. We can pair up these factors to form perfect cubes, such as 1 and 27, 3 and 9, and so on. Each of these pairs results in a perfect cube, which means that the cube root of the fraction will be a whole number.
Conclusion
In conclusion, understanding the properties of cube roots and perfect cubes is essential in solving this problem. By identifying the perfect cubes that can be formed using the factors of 144, we can determine the number of different values of $y$ for which the expression $\sqrt[3]{\frac{144}{y}}$ results in a whole number.