Exponential Equations Pg 33 Q12

[Guest]

Given that x is an integer greater than 1, how can you determine whether the following expression CAN be an integer ?

X^1/4 + x^1/2

whats the best way to answer this ?

Thanks

[StaceyKoprince]

x^(1/4) means to take the 4th root
x^(1/2) means to take the square root
I'll assume you are familiar with the above and don't need an explanatio there.

If I'm given that x is an integer and I want that sum to be an integer, what I really want is for each term in the sum to be an integer: x^(1/4) = integer, and x^(1/2) = integer.

The limiting item here is the first one: x^(1/4). If some number is a perfect square (that is, x^(1/2) = integer), the 4th root may or may not be an integer. If, however, some number represents an integer raised to the fourth power, then x^(1/4) = integer AND x^(1/2) = integer.

So we'd want to be able to determine whether the number x is an integer raised to the 4th power - if so, the expression represents an integer. If the question is merely "can this be an integer?" ask yourself "can I think of an integer that I can then raise to the 4th power - and that will be x?" Of course, you can, so the answer is yes, it can be an integer.