If y is a positive integer, for how many different values of y is sqrt ^ 3 144 / y a whole number?

Answer: second option.Step-by-step explanation: We need the remember that: [tex]\sqrt[n]{a^n}=a[/tex] Therefore, the radicand (In this case [tex]\frac{144}{x}[/tex]) must be a perfect cube in order to get a whole number. Remember that a perfect cube are the numbers that have exact cube roots. Therefore, "y" must be a factor of 144. Now, we need to descompose 144 into its prime factors: [tex]144=2*2*2*2*3*3=2^4*3^2[/tex] Then, the factors of 144 are: [tex]1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, 144[/tex] Substitute values into [tex]\frac{144}{x}[/tex] to check: [tex]\frac{144}{1}=144[/tex] [tex]\frac{144}{2}=72[/tex] [tex]\frac{144}{3}=48[/tex] [tex]\frac{144}{4}=36[/tex] [tex]\frac{144}{6}=24[/tex] [tex]\frac{144}{8}=18[/tex] [tex]\frac{144}{9}=16[/tex] [tex]\frac{144}{12}=12[/tex] [tex]\frac{144}{16}=9[/tex] [tex]\frac{144}{18}=8[/tex]  (Perfect cube: [tex]8=2^3[/tex]) [tex]\frac{144}{24}=6[/tex] [tex]\frac{144}{36}=4[/tex] [tex]\frac{144}{48}=3[/tex] [tex]\frac{144}{72}=2[/tex] [tex]\frac{144}{144}=1[/tex] (Perfect cube: [tex]1=1^3[/tex]) Therefore, for 2 different values of "y"  ([tex]y=18[/tex] and [tex]y=144[/tex]) [tex]\sqrt[3]{\frac{144}{y} }[/tex] is a whole number.