Department of Mathematics --- College of Science --- University of Utah

Mathematics 1010 online

Roots and Radicals

Roots and Radicals deserve their own chapter and homework because they occur frequently in applications.

Let $ n\geq 2 $ be a natural number , and let $ a $ be a real number . The $ n $-th root of $ a $ is a number $ b $ that satisfies $ a = b^n. $ The number $ b $ is denoted by

$\displaystyle b = \root n \of a. $

For example, $ \root 4 \of 81 = 3 $ since $ 3^4 = 81 $, and $ \root 5 \of 32 = 2 $ since $ 2^5= 32 $.

The symbol $ \sqrt{\phantom{yz}} $ is called the radical symbol, and an expression involving it is called a radical (expression).

If $ n=2 $ then $ b $ is the square root of $ a $ and the number $ 2 $ is usually omitted. For example, $ \root 2 \of 25 = \sqrt{25} = 5. $

If $ n=3 $, then $ b $ is the cube root of $ a $. For example, the cube root of $ 27 $ is $ 3 $, and that of $ 8 $ is $ 2 $.

If $ n $ is even and $ a $ is positive then there are two $ n $-th roots of $ a $, each being the negative of the other. For example, since $ 5^2 = (-5)^2 = 25 $ there are two square roots of $ 25 $. In that case by convention the symbol $ \root n \of a $ means the positive $ n $-th root of $ a $, and it is called the principal ($ n $-th) root of $ a $.

If $ a $ is negative and $ n $ is odd then there is just one $ n $-th root, and it is negative also. For example, $ \root 3 \of -1 = \root 5 \of -1 = -1. $

At this stage we do not know of an $ n $-th root if $ n $ is even and $ a $ is negative. This leads to the subject of complex numbers which we will take up later in the course.

Radicals are just special cases of powers, and you can simplify much of your thinking by keeping this fact in mind:

$\displaystyle \root n \of a = a^{\frac{1}{n}}. $

It follows immediately from that observation and the properties of powers that

$\displaystyle a^\frac{m}{n} = a^{m\times\frac{1}{n}} = \root n \of {\left(a^m\right)} = \left(\root n \of a\right)^m. $

Solving Radical Equations

An equation involving radicals is called a radical equation (naturally). To solve it you simply apply our general principle:

To solve an equation figure out what bothers you and then do the same thing on both sides of the equation to get rid of it.

To get rid of a radical you take it to a power that will change the rational exponent to a natural number. This will work if the radical is on one side of the equation by itself.

Let's look at a few simple examples:

Suppose

$\displaystyle \root 3 \of x -5 = 0. $

We proceed as follows:

\begin{displaymath} \begin{array}{rclcl} \root 3 \of x -5 &=& 0 &\vert& +5 \\ ... ...()^3 \\ x &=& 125 &\vert& \hbox{the answer}.\\ \end{array} \end{displaymath}

Here is a slightly more complicated problem:

$\displaystyle \root 5 \of {2x+2} -1 = 0. $

We obtain

\begin{displaymath} \begin{array}{rclcl} \root 5 \of {2x+2} -1 &=& 0 &\vert& + 1... ... x &=& - \frac{1}{2} &\vert& \hbox{the answer}\\ \end{array} \end{displaymath}

Our last example shows how to get rid of more than one radical:

$\displaystyle \sqrt{x} + \sqrt{x-1} = 3. $

To get rid of the square roots we isolate them and square one at a time:

\begin{displaymath} \begin{array}{rclcl} \sqrt{x} + \sqrt{x-1} &=& 3 &\vert& - ... ... x &=& \frac{25}{9} &\vert& \hbox{the answer} \\ \end{array} \end{displaymath}

In each case, we check our answer by substituting it in the original equation. For example, in the last equation we obtain:

$\displaystyle \sqrt{x} + \sqrt{x-1} = \sqrt{\frac{25}{9}} + \sqrt{\frac{16}{9}} = \frac{5}{3} + \frac{4}{3} = 1. $

Later in the course we will consider more complicated cases of radical equations.

Numerical Values

The radicals in the above examples were all natural numbers. This is due only to a judicious choice of examples. Frequently the roots occurring in applications are irrational numbers with decimal expansions that never repeat or terminate. The following table lists approximations of a few specific radicals.

\begin{displaymath} \begin{array}{rccccc} a & \sqrt{a} & \root 3 \of a & \root ... ....16228 & 2.15443 & 1.77828 & 1.58489 & 1.46780\\ \end{array} \end{displaymath}

Some Radicals (Approximately)