if (y) is a positive integer, for how many different values of (y) is (sqrt3{\frac{144}{y}}) a whole…

Answer:

A. 1

Explanation:

Step1: Prime - factorize 144

$144 = 2^{4}\times3^{2}$

Step2: Analyze cube - root condition

For $\sqrt[3]{\frac{144}{y}}$ to be a whole number, $y$ must be such that when 144 is divided by $y$, the result is a perfect - cube. The prime - factorization of a perfect cube has exponents that are multiples of 3. The exponents of 2 and 3 in the prime - factorization of 144 are 4 and 2 respectively. There is no positive integer $y$ for which $\frac{144}{y}$ is a perfect cube. So the number of positive - integer values of $y$ for which $\sqrt[3]{\frac{144}{y}}$ is a whole number is 1 (when $y = 144$, $\sqrt[3]{\frac{144}{144}}=\sqrt[3]{1}=1$).