Final answer:
A square root over a number is rational if the square root can be expressed as a fraction of two integers, whereas it is irrational if it cannot. For example, √16 is rational because it equals 4, which is 4/1, but √2 is irrational as it cannot be simplified to a fraction of integers.
Step-by-step explanation:
The question asks whether a square root over a number would still be considered rational. A rational number is one that can be expressed as the fraction of two integers, where the denominator is not zero. While all fractions with integer numerators and denominators are rational, not all square roots are. For example, the square root of a perfect square, such as √16 (which is 4), is rational because it can be expressed as a fraction (4/1). However, the square root of a number that is not a perfect square, such as √2, cannot be expressed as a fraction of two integers, and therefore, it is irrational.
It is important to note that if the square root of a number can be simplified to a fraction of two integers, then it is rational. The exponents and roots are closely related in mathematics. For instance, x² = √x can be seen as an exponentiation. The square root of x can be re-expressed as x raised to the power of 1/2, and if the result is a fraction of two integers, it remains rational.