Algebra Know-It-ALL

exponentsp and q be any rational numbers we want. That gives us the powerful, far-reaching
generalized multiplication-of-exponents (GMOE) rule! If a, p, and q are rational numbers and
a≠ 0, then

(ap)q=apq

We now have a way to evaluate an expression where we raise a number to a certain power, and
then take a root of the result. Remember that a root is a reciprocal power. So, if we encounter
an exponent that takes the form r/s, we can call this the product of r and 1/s, and then use
the GMOE rule:

(ar)1/s=ar(1/s)=ar/s

That’s how we’d evaluate the sth root of ar. But it also tells us something more: when we take
a base number to a rational-number, noninteger power, it’s the same thing as taking the base
to an integer power and then taking an integer root of the result. Remember, a rational num-
ber is a quotient of two integers! If we reverse the order of the terms in the above three-way
equation, we get

ar/s=ar(1/s)= (ar)1/s

This is a heavy dose of abstract math! Let’s look at a couple of specific cases where integers are
raised to rational-number powers. First, this:

10 6/3= 10 6×(1/3)

= (10^6 )1/3

= 1,000,0001/3

= 100

If you’re astute, you can solve this a lot quicker by noting that 6/3 = 2, so

10 6/3= 102

= 100

Usually, rational-number powers aren’t this easy to evaluate. The results often produce num-
bers that aren’t even rational. Consider this example:

2 3/2= 23 ×(1/2)

= (2^3 )1/2

= 8 1/2

= 2.8284 ...

This is an endless nonrepeating decimal. It cannot be expressed as a ratio of integers, and is not
a rational number. You’ll learn more about these types of numbers in the next chapter.

Multiple Powers 121