prove that the square root and cube root of every positive i

prove that the square root and cube root of every positive integer is an algebraic number.

Solution

An algebraic number is a possibly complex number that is a root of a finite, non-zero polynomial in one variable with rational coefficients (or equivalently – by clearing denominators – with integer coefficients).

Now first consider square root of any number

If the integer is a perfect square we get rational number as square root otherwise an irrational number

Thus every integer has square root as algebraic number

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For cube root:

If perfect cube then it is a rational number otherwise rational real number for cube root