Exponents with negative bases
A basic topic when it comes to exponents is the fact that when you are raising a negative exponent to a
power, an even power will produce a positive answer and an odd power will produce a negative answer. This
is illustrated by the following:
(-2)3=(-2)·(-2)·(-2)=-8
(-2)4=(-2)·(-2)·(-2)·(-2)=16
Whenever you are multiplying a long string of negative and positive numbers, the easiest way to determine
the overall sign i sto count the number of negatives involved in the product.
 If the product involves an even number of negatives, the overall sign of the product will be positive.
 If the product involves an odd number of negatives, the overall sign of the product will be negative.
Will the following products be positive or negative?
a. (-4)8 b. (-5)3
c. -3·(-6)8
d. -2·33 e. 3·(-2)4 f. -4·(-3)3
a. The overall product will be positive. There is an even number (8) of negative signs in
the product.
b. The overall product will be negative. There is an odd number (3) of negative signs in
the product.
c. The overall product will be negative: There are 9 negative signs in the product (one
from the -3 and and 8 from (-6)8.
d. The overall will be negative. There is only one negative sign in the product.
e. The overall product will be positive. There are 4 negative signs involved in the
product (from (-2)4).
f. The overall product will be positive: There are 4 negative signs involved in the
product (one from the -4 and 3 from (-3)3).
When combining negative numbers and exponents, parentheses become very important:
Compare (-2)4 and -24
(-2)4=(-2)·(-2)·(-2)·(-2)=16
-24=-2·2·2·2=-16