Algebra Know-It-ALL

118 Powers and Roots

Let’s call this the addition-of-exponents (AOE) rule. We can see how it works by trying out an

example with specific numbers. Let a= 3, m= 2, and n=−4. Then

32 × 3 −^4 = 9 × 1/81

= 9/81

= 1/9

and

3 [2+(−4)]= 3 (2−4)

= 3 −^2

= 1/(3^2 )

= 1/9

Divide by subtracting

Think of a nonzero number, b, along with two quantities bp and bq, where p and q are integers.

If you divide the first of these quantities by the second, you get the same result as if you sub-

tractq from p, and then raise b to that power. Mathematically:

bp/bq=b(p−q)

Let’s call this the subtraction-of-exponents (SOE) rule. Now we’ll work out an example. Suppose

we have b= 10, p= 5, and q= 3. Then

105 /10^3 = 100,000/1,000

= 100

and

10 (5−3)= 102

= 100

Are you confused (yet)?

Was that too easy for you? Let’s try a slightly tougher example. Let b=−2,p= 3, and q= 4. Then

(−2)^3 /(−2)^4 =−8/16

=−1/2

and

(−2)(3−4)= (−2)−^1

= 1/(−2)

=−1/2