We will shed some light on nth roots and rational exponents of numbers. We give a clear and rigorous definition of these root numbers. Examples and solved exercises are also given.
Generality on nth roots
We explain in a simple manner what is the nth roots of a number. We also deal with the rational exponents of numbers.
Square roots numbers and more roots
Although the square roots are standard and they are very restrictive. In fact, a number is the square root of a positive number if . In this case, we write sometimes we denote . Thus square roots are solutions of equations of type .
Naturel question: Does exists numbers positive integer number and a real number such that is not positive, i.e. negative? Yes, we can find examples. In fact, . Then is the solution of the equation . We say that is the cubic root of the number and we write . So unlike square roots, cubic roots can be negative.
More generally, let be a non-zero integer and a number, not necessarily an integer, and ask the question: Are their numbers such that . Nice that . Thus necessarily is positive and in this case is called -th root of and we write or . If is negative, then there is no solution.
Thus for even numbers of the form , the nth root must be positive.
Now assume that we have an odd number and look for numbers satisfying the algebraic equation . This is, equivalent to . As is positive, it follows that the numbers and have the same sign, either both positive or negative. In this case, the solution exists. Now if and have opposite signs, then there is no solution to the above equation.
In order to be more precise on the nth roots, let the following definition of the components of a radical expression
If we set . Then we have the following cases: If the index is even: then the radicant and must be positive. If index is an odd number: then either and are positive or and are negative.
Example: take am index odd number, , the radicand is negartive.
Rational exponents
A fraction exponent, also called fractional exponent, of a number is given by where and are relative numbers. Simply the name rational means that the exponent of the number is a fraction of the forme .