To determine for how many different values of [tex]\( y \)[/tex] the expression [tex]\( \sqrt[3]{\frac{144}{y}} \)[/tex] is a whole number, we need to follow these steps:
1. Understanding the condition: The expression [tex]\( \sqrt[3]{\frac{144}{y}} \)[/tex] should result in a whole number, which means that [tex]\(\frac{144}{y}\)[/tex] must be a perfect cube.
2. Finding the perfect cubes: To find such [tex]\( y \)[/tex], we need to identify all divisors of 144 that make [tex]\(\frac{144}{y}\)[/tex] a perfect cube.
3. Factorizing 144: First, factorize 144 into its prime factors:
[tex]\[
144 = 12 \times 12 = (2^2 \cdot 3) \times (2^2 \cdot 3) = 2^4 \times 3^2
\][/tex]
4. Looking for perfect cubes: For [tex]\(\frac{144}{y}\)[/tex] to be a perfect cube, [tex]\( \frac{144}{y} \)[/tex] should take values that are cubes of integers. Given the prime factorization
[tex]\[
144 = 2^4 \times 3^2
\][/tex]
we recognize that if we want [tex]\(\frac{144}{y}\)[/tex] to be [tex]\( k^3 \)[/tex] for some integer [tex]\( k \)[/tex], then [tex]\( 2^4 \times 3^2 \div y = k^3 \)[/tex].
5. Identifying possible values of [tex]\( y \)[/tex]:
By examining the possible perfect cubes less than or equal to 144, we note:
- [tex]\( k = 1 \Rightarrow k^3 = 1 \Rightarrow \frac{144}{y} = 1 \Rightarrow y = 144 \)[/tex]
- [tex]\( k = 2 \Rightarrow k^3 = 8 \Rightarrow \frac{144}{y} = 8 \Rightarrow y = 18 \)[/tex]
- Higher values [tex]\( k \geq 3 \)[/tex] lead to [tex]\( k^3 \)[/tex] greater than 144.
Thus the positive integer values of [tex]\( y \)[/tex] that make [tex]\(\sqrt[3]{\frac{144}{y}}\)[/tex] a whole number are [tex]\( y = 18 \)[/tex] and [tex]\( y = 144 \)[/tex].
6. Conclusion: There are exactly two positive integer values for [tex]\( y \)[/tex] that satisfy the given condition: [tex]\( \boxed{2} \)[/tex].
