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

Let be a
natural number , and let be a
real number . The **-th root of **
is a number that satisfies
The number is denoted by

For example, since , and since .

The symbol
is called the **radical symbol,** and
an expression involving it is called a **radical (expression)**.

If then is the** square root** of and the number is usually omitted. For example,

If , then is the **cube root** of . For example,
the cube root of is , and that of is .

If is even and is positive then there are two -th
roots of , each being the negative of the other. For example,
since
there are two square roots of . In
that case by convention the symbol
means the positive
-th root of , and it is called the **principal (-th) root
of . **

If is negative and is odd then there is just one -th root, and it is negative also. For example,

At this stage we do not know of an -th root if is even and 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:

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

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

Here is a slightly more complicated problem:

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

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

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

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

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.