Question on Non Negative Real Values

Topic : Non Negative real Numbers
Question : For how many non-negative real values of x is √[144 - ³√(x)] an integer?

Solution :
The root of an integer that is not a square number cannot be an integer.

It is given that x is a non-negative real number. So x takes only the positive values.

Therefore the values within square root will be the square numbers less than or equal to 144 so that √[144 - ³√(x)] is an integer.

The possible values within the square are :
144, 121, 100, 81, 64, 49, 36, 25, 16, 9, 4, 1, 0

The possible values of ³√(x) are:
(144-144), (144-121), (144-100), (144-81), (144-64), (144-49), (144-36), (144-25), (144-16), (144-9), (144-4), (144-1), (144-0)

0, 23, 44, 63, 80, 95, 108, 119, 128, 135, 140, 143, 144

Therefore the possible non-negative real values of x for which √[144 - ³√(x)] an integer are:
0, 23³, 44³, 63³, 80³, 95³, 108³, 119³, 128³, 135³, 140³, 143³, 144³

Therefore for 13 non-negative real values of x, √[144 - ³√(x)] is an integer.