I am tired of fractions, you too? Well let’s switch gears and talk about the roots of numbers. This is preparing you for a post or posts on the rules of exponents.

So before, I introduced the concept of the square root and how it is the reverse operation of squaring a number:

\[

\sqrt{25}\hspace{0.33em}{=}\hspace{0.33em}\pm{5}

\] because \[

{5}^{2}\hspace{0.33em}{=}\hspace{0.33em}{25}

\]. That is \[

\sqrt{{5}^{2}}\hspace{0.33em}{=}\hspace{0.33em}{5}

\]. Or in general, \[

\sqrt{{x}^{2}}\hspace{0.33em}{=}\hspace{0.33em}\pm{x}

\]. Remember that when taking the square root, there are two solutions since \[

{\left({{-}{5}}\right)}^{2}\hspace{0.33em}{=}\hspace{0.33em}{25}

\] as well.

Well, what about the opposite operation to \[

{x}^{3}

\]? Well there is one:

\sqrt[3]{{x}^{3}}\hspace{0.33em}{=}\hspace{0.33em}{x}

\]

Notice a few things here. First, there is only one solution. There are no plus/minus solutions because the index (the “3”) of the root is odd and using the rules of signs, the sign of the odd root of a number will be the same as the number in the radical symbol. That is:

\[\sqrt[3]{125}\hspace{0.33em}{=}\hspace{0.33em}{5}{,}\hspace{0.33em}\hspace{0.33em}\sqrt[3]{{-}{125}}\hspace{0.33em}{=}\hspace{0.33em}{-}{5}

\]

The other thing to notice is that if there is no index shown, then a “2” is assumed to be there. If the index is a “3”, it is called a *cube root. *After that, you use ordinal numbers, that is fourth root, fifth root, etc.

The last thing to notice is if the index is even, then you do get two plus/minus solutions. If the index is odd, you only get one solution.

So in general,

\[

\sqrt[n]{{x}^{n}}\hspace{0.33em}{=}\hspace{0.33em}\pm{x}

\] if *n* is even and

\[

\sqrt[n]{{x}^{n}}\hspace{0.33em}{=}\hspace{0.33em}{x}

\] if *n* is odd.

Examples:

\[\begin{array}{l}

{\sqrt[4]{16}\hspace{0.33em}{=}\hspace{0.33em}\sqrt[4]{{(}\pm{2}{)}^{4}}\hspace{0.33em}{=}\hspace{0.33em}\pm{2}}\\

{\sqrt[3]{{-}{8}}\hspace{0.33em}{=}\hspace{0.33em}\sqrt[3]{{(}{-}{2}{)}^{3}}\hspace{0.33em}{=}\hspace{0.33em}{-}{8}}\\

{\sqrt[5]{32}\hspace{0.33em}{=}\hspace{0.33em}\sqrt[5]{{2}^{5}}\hspace{0.33em}{=}\hspace{0.33em}{2}}\\

{\sqrt[5]{{-}{32}}\hspace{0.33em}{=}\hspace{0.33em}\sqrt[5]{{(}{-}{2}{)}^{5}}\hspace{0.33em}{=}\hspace{0.33em}{-}{2}}\\

{\sqrt[4]{81}\hspace{0.33em}{=}\hspace{0.33em}\sqrt[4]{{(}\pm{3}{)}^{4}}\hspace{0.33em}{=}\hspace{0.33em}\pm{3}}

\end{array}

\]

In my next post, I’ll introduce some rules regarding exponents.