Algebra Know-It-ALL

114 Powers and Roots

The third power is often called the cube. We can say, “4 cubed equals 64.” Now if we go with

the reciprocal power and work backwards, we get

64 1/3= 4

This can be read as, “The cube root of 64 equals 4.”

The 3rd power is called the cube and the 1/3 power is called the cube root because of the

relationship between the edges and the interior volume of a geometric cube. For any perfect

cube, the volume is equal to the 3rd power of the length of any edge (Fig. 8-2). Going the

other way, the length of any edge is equal to the 1/3 power of the volume. The figure also

shows the radical notation for the cube root. The fact that the radical refers to the cube root,

rather than the square root, is indicated by the small numeral 3 in the upper-left part of the

radical symbol.

Higher roots

Whenp is a positive integer equal to 4 or more, people write or talk about the numerical pow-

ers and roots directly. That’s because geometric hypercubes having 4 dimensions or more are not

commonly named. A 4-dimensional hypercube is technically called a tesseract, but you should

expect incredulous stares from your listeners if you say “2 tesseracted is 16” or “The tesseract

root of 81 is 3.”

Here are some examples of higher powers and roots. You can check the larger ones on

your calculator if you like.

24 = 16 so 161/4= 2

34 = 81 so 811/4= 4

Length of edge = s

Length of edge =

s

Interior

volume = V

Length of

edge = s

=s

s=

and

V

V

3

1/3

= V

3

Figure 8-2 The volume of a geometric cube is equal to the

3rd power, or cube, of the length of any edge.

Therefore, the length of any edge is equal to

the 1/3 power, or cube root, of the volume.