Algebra Know-It-ALL

122 Powers and Roots

Are you confused?

Let’s review the most important points in this chapter. They can be condensed into six statements.

• If a is any nonzero number and −n is a negative integer, the expression a−n means you should raise
  a to the power of |n|, and then take the reciprocal of the result.


• If a is any nonzero number and m and n are rational numbers, then am times an is the same as a
  raised to the power of (m+n).


• If a is any nonzero number and m and n are rational numbers, then am divided by an is the same
  asa raised to the power of (m−n).


• If a is any nonzero number and p is any nonzero integer, then the pth root of a is the same as rais-
  inga to the power of 1/p.


• If a, p, and q are rational numbers and a is nonzero, then if you raise a to the pth power and take
  the result to the qth power, it’s the same as raising a to the power of pq.


• If a, p, and q are rational numbers with a and q nonzero, then if you raise a to the pth power and take
  theqth root of the result, it’s the same as raising a to the power of p/q.

Here’s a challenge!

What do you get if you take the −5/2 power of 6? Mathematically, evaluate this expression and use a cal-

culator to figure out the result to several decimal places:

6 (−5/2)

Solution

Let’s apply the GMOE rule to this problem. It can be tricky because of the minus sign, and we have to be

sure we remember the difference between negative powers and reciprocal powers. Let’s go:

6 (−5/2)= 6 −^5 ×(1/2)

= (6−^5 )1/2

= [1/(6^5 )]1/2

= (1/7,776)1/2

Now it’s time to use a calculator! Remember that the 1/2 power is the same as the square root. First we take

the reciprocal of 7,776, getting a decimal point, three ciphers, and a long string of digits. Then we hit the

square root key with the string of digits still in the display, getting

0.01134023 ...

The digits go on without end, and there’s no apparent pattern. As things turn out, this is not a rational

number.