Question

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

Answer 1

B) 2

Explanation:

Edge 2020

Answer 2

Answer: second option.

Explanation:

We need the remember that:

`\sqrt[n]{a^n}=a`

Therefore, the radicand (In this case
`(144)/(x)`) 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:

`144=2*2*2*2*3*3=2^4*3^2`

Then, the factors of 144 are:
`1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, 144`

Substitute values into
`(144)/(x)` to check:

`(144)/(1)=144`

`(144)/(2)=72`

`(144)/(3)=48`

`(144)/(4)=36`

`(144)/(6)=24`

`(144)/(8)=18`

`(144)/(9)=16`

`(144)/(12)=12`

`(144)/(16)=9`

`(144)/(18)=8` (Perfect cube:
`8=2^3`)

`(144)/(24)=6`

`(144)/(36)=4`

`(144)/(48)=3`

`(144)/(72)=2`

`(144)/(144)=1` (Perfect cube:
`1=1^3`)

Therefore, for 2 different values of "y" (
`y=18` and
`y=144`)
`\sqrt[3]{(144)/(y) }` is a whole number.