What does it mean for expressions to be non-integral and non-rational in a variable?

[Drain Brain]

can you explain what it means when they are not "Integral" and not "Rational"?

$\frac{4y}{x} = 4yx^{-1 }$ is not integral in x
$3x\sqrt{y}z^3$ not rational in y

 

[Prove It]

can you explain what it means when they are not "Integral" and not "Rational"?

$\frac{4y}{x} = 4yx^{-1 }$ is not integral in x
$3x\sqrt{y}z^3$ not rational in y

To be integral, your expression needs to always give an integer value. If x is anything but 1 or a common factor of the top, then your first expression will not be integral.

To be rational, your expression needs to always give a rational (fractional) value. If y is anything but a perfect square of a rational number, then the second expression will not be rational.