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Exponents


Exponents afford a convenient way of expressing long products of the same number. The expression bn is called a power and it stands for b × b × b × • • • × b, where there are n factors of b. b is called the base, and n is called the exponent. By definition, b° = 1.

 

There are six rules that govern the behavior of exponents:


Rule 1:
xaxb = xa + b

Example:
23 • 22 = 23+2 = 25 = 32.
Caution:
xa + xb x a+ b


Rule 2:
( xa )b = xab
Example:
(2 3)2= 23•2 = 26 = 64


Rule 3:
(x y)a = x a • ya
Example:
(2y)3 = 23y3 = 8y3


Rule 4:
x y a = x a y a
Example:
x32 = x232 = x29


Rule 5:
x a x b = x a - b , if a > b
Example:
2623 = 26-3 = 23 = 8
xaxb = 1xb-a, if b>a.
Example:
2326 = 126-3 = 123 = 18


Rule 6:
x - a = 1 x a
Example:
z-3 = 1z3 Caution, a negative exponent does not make the number negative; it merely indicates that the base should be reciprocated. For example, 3-2 ≠ - 132 or - 19.

Problems involving these six rules are common on the GRE, and they are often listed as hard problems. However, the process of solving these problems is quite mechanical: simply apply the six rules until they can no longer be applied.

 

Example 1:
If x ≠ 0, xx52x4 = ?

(A) x 5
(B) x 6
(C) x 7
(D) x 8
(E) x 9
 

First, apply the rule ( x a )b = xab to the expression xx52x4 :
x·x5·2x4 = x·x10x4

Next, apply the rule xaxb = xa +b :

x·x10x4 = x11x4

Finally, apply the rule xaxb = xa-b :

x11x4 = x11-4 = x7

The answer is (C).
 
Note: Typically, there are many ways of solving these types of problems. For this example, we could have begun with Rule 5, xaxb = 1xb-a :
xx52x4 = xx52x 4-1 = xx52x3

Then apply Rule 2,(xa)b = xab:

x52x3 = x10x3

Finally, apply the other version of Rule 5, xaxb = xa - b:

x10x3 = x7

 

 

Example 2:
 

Column A Column B
3·3·3·39·9·9·9 134
Canceling the common factor 3 in Column A yields 1·1·1·13·3·3·3, or 13131313 . Now, by the definition of a power, 13131313 = 134 Hence, the columns are equal and the answer is (C).

 

 

Example 3:

Column A
6432

Column B
24•32
 

First, factor Column A:
2·3432

Next, apply the rule (x y)a = x a • ya :
24·3432

Finally, apply the rule xaxb = xa - b:

24•32

Hence, the columns are equal and the answer is (C).




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