Example 1

You are probably familiar with expressions such as and . Both are examples of radicals. They are radicals because of the radical sign, . But it is improper to verbalize as “radical 9.” That would be ambiguous because the type of radical, square root, has not beeen identified. It is the same with . Would you say “radical 64” when someone might think you are talking about the square root of 64 instead of the cube root or some other root?

The term ”square root” comes from the fact that a square root is in some sense the “opposite” of squaring a number. Since squaring a number would give you the area of a square, the square root of the area of a square is the length of a side of that square.

Below is a square with an area of 9 square units. (Here a unit is the length of the side of one of the little boxes.) So the square root of 9 is 3 since the length of the side of the larger square is 3 units. But you knew that, right?

The ancient Greeks and others thought about square roots and cube roots in terms of geometry such as this. Of course, cube roots are related to cubes. I do not know if they broadened their perspective by considering other kinds of roots. It would have been natural for them to dismiss something like a twelfth root of 2, , as ridiculously abstract, of a twelfth dimensional character non-existent in our three dimensional world. How ironic since the use of this number to tune keyboard instruments since around 1720 superceded the note tuning system devised by Pythagoras and others!

We can come up with an algebraic way of defining a square root in order to justify .

There is a small problem with this type of definition.

Does this mean that there are two square roots of 9? Yes. Does this mean that No. In modern terms, we define any radical as a function. You may recall from Section 2.1 that a function has a single output. We choose the output for a square root to be a positive number (or zero, in the case of the square root of zero). This positive square root is known as the principle square root. The negative square root of 9 is - 3. Here is a summary.

To generalize this definition of square root textbooks write something like this.

The absolute value of a number will be positive (or zero in the case of zero), so this definition is consistent with providing both a single output to the square root function that is positive, and with providing a way to identify the negative square root.

The cube root of 64, , is 4, since . This would correspond to a cube having a volume of 64 cubic units. Each side would have length 4.

Just as numbers can have two square roots, numbers can also have three cube roots. The other two cube roots exist in a different numerical world, the complex numbers, which will be introduced in Section 7.7. To find these, however, is difficult without the use of trigonometry. All that is beyond the scope of this course.

The little 3 written in designates the type of radical and is known as the index of the radical. Sometimes we might call a “third root” of 64. Indices of radicals normally are positive whole numbers. Thus the index of is 12, so we say this is the “twelfth root of 2.”

The index of a square root is 2. We could write something like as , but there is seldom any reason to do so. It is customary to interpret a radical sign without an index as a square root, or second root.

Incidentally, .

Copyright © 2005 by Jon Davidson. Duplication for instructional usage is permitted for students and faculty of Southern State Community College.