Nth roots and rational exponents

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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 a is the square root of a positive number b if a2=b. In this case, we write a=b, sometimes we denote a=b12. Thus square roots are solutions of equations of type a2=b.

Naturel question: Does exists numbers m positive integer number and a real number x such that xm is not positive, i.e. negative? Yes, we can find examples. In fact, (3)3=(3)×(3)×(3)=9. Then x=3 is the solution of the equation x3=9. We say that 3 is the cubic root of the number 9 and we write 3=93. So unlike square roots, cubic roots can be negative.

More generally, let k be a non-zero integer and b a number, not necessarily an integer, and ask the question: Are their numbers x such that x2k=b. Nice that x2k=(xk)2. Thus necessarily b is positive and in this case x is called 2k-th root of b, and we write x=b2k or x=b12k. If b is negative, then there is no solution. 

Thus for even numbers n, of the form n=2k, the nth root must be positive. 

Now assume that we have an odd number n=2k+1, and look for numbers x satisfying the algebraic equation xn=b. This is, equivalent to x2kx=b. As x2k is positive, it follows that the numbers x and b have the same sign, either both positive or negative. In this case, the solution exists. Now if x and b 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

 

components-of-a-radical-expression


If we set b=xn. Then we have the following cases
: If the index n is even: then the radicant x and b must be positive. If index n is an odd number: then either x and b are positive or x and b are negative.

Example: take am index n=3, odd number, b=9, the radicand x=3 is negartive.

Rational exponents

A fraction exponent, also called fractional exponent, of a number a is given by amn=(an)m=amn where n and m are relative numbers. Simply the name rational means that the exponent of the number a is a fraction of the forme mn.

A selection of exercises on nth roots

Exercise: Simplify the following expressions ((33)24)6,32n+1n,543523.

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