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Nth roots and rational exponents


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 $a^2=b$. In this case, we write $a=\sqrt{b},$ sometimes we denote $a=b^{\frac{1}{2}}$. Thus square roots are solutions of equations of type $a^2=b$.

Naturel question: Does exists numbers $m$ positive integer number and a real number $x$ such that $x^m$ is not positive, i.e. negative? Yes, we can find examples. In fact, $(-3)^3=(-3)\times (-3)\times (-3)=-9$. Then $x=-3$ is the solution of the equation $x^3=-9$. We say that $-3$ is the cubic root of the number $-9$ and we write $-3=\sqrt[3]{-9}$. 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 $x^{2k}=b$. Nice that $x^{2k}=(x^k)^2$. Thus necessarily $b$ is positive and in this case $x$ is called $2k$-th root of $b,$ and we write $x=\sqrt[2k]{b}$ or $x=b^{\frac{1}{2k}}$. 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 $x^n=b$. This is, equivalent to $x^{2k} x =b$. As $x^{2k}$ 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



If we set $b=\sqrt[n]{x}$. 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 $$  \sqrt[n]{a^m}=\left( \sqrt[n]{a}\right)^m=a^{\frac{m}{n}}$$ 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 $\frac{m}{n}$.

A selection of exercises on nth roots

Exercise: Simplify the following expressions \begin{align*} \left(\sqrt[4]{\left(\sqrt[3]{3} \right)^2} \right)^6,\quad \sqrt[n]{3^{2n+1}},\quad \frac{5^{4}{3}}{5^{-\frac{2}{3}}}.\end{align*}

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