Algebra Know-It-ALL

120 Powers and Roots

When exponents multiply

Imagine that we have a number a that is not equal to 0. Suppose p and q are integers. What

happens if we raise a to the pth power, and then raise the result to the qth power? Mathemati-

cally, we get this expression:

(ap)q

It is tempting to suppose that the result of this operation will always produce a huge number.

That can happen if the absolute value of a is larger than 1, and if p and q are both positive and

more than 1. For example:

(4^5 )^6 = 1,024^6

= 1,152,921,504,606,846,976

It doesn’t always work out that way, however. If the absolute value of a is between 1 and 0,

and if p and q are both positive and more than 1, the number may be quite close to 0. For

example,

[(−0.1)^3 ]^5 = (−0.001)^5

=−0.000000000000001

When you have any expression of this sort, you can get the same result if you take the base a

to the power of the product of the exponents pq. That is,

(ap)q=apq

Let’s call this the multiplication-of-exponents (MOE) rule. Looking at the numerical examples

we just saw, and putting them in this form, illustrates this:

(4^5 )^6 = 45 ×^6

= 430

= 1,152,921,504,606,846,976

and

[(−0.1)^3 ]^5 = (−0.1)^3 ×^5

= (−0.1)^15

=−0.000000000000001

You can evaluate expressions with large exponents quickly by using a calculator with an “x to

theyth power” key.

Rational-number powers

When you raise an exponentiated quantity (i.e., anything to a power) to another power, either

or both of the exponents can be negative. The MOE rule still applies. In fact, we can let the