Algebra Know-It-ALL

The second power is often called the square, so we can say, “5 squared equals 25.” By defini-
tion then

25 1/2= 5

We would say, “The square root of 25 is equal to 5.” In general, for any two numbers a and b,
and for any positive integer p, we can say this:

If ap=b, then b1/p=a

The reason the 2nd power is called the square and the 1/2 power is called the square root can
be explained in terms of the dimensions and area of a perfect geometric square. For any geomet-
ric square, the interior area is equal to the 2nd power of the length of any one of the edges, as
shown in Fig. 8-1. That’s why the 2nd power is called the square. Looking at it the other way,
the length of any one of the edges is equal to the 1/2 power of the interior area. That’s why the
1/2 power is called the square root. In the figure, the radical notation for square root is shown,
in addition to the 1/2 power notation. The radical consists of a surd symbol (√) with a line
extending over the top of the quantity of which the square root is taken.

The cube root

Now let’s see what happens when p= 3, so 1/p= 1/3. We can easily figure out what happens
when we raise a number, say 4, to the 3rd power:

43 = 4 × 4 × 4 = 64

Interior

area = A

Length of edge = s

Length of edge =

s

A=s^2

s=A1/2

and

= A

Figure 8-1 The area of a geometric square is equal to

the 2nd power, or square, of the length of

any edge. Therefore, the length of any edge

is equal to the 1/2 power, or square root,

of the area.

Reciprocal-of-Integer Powers 113