3.2. Properties of Exponents http://www.ck12.org
(a−^2 b^3 )−^3
ab^2 c^0Solution:
(a−^2 b^3 )−^3
ab^2 c^0 =a^6 b−^9
ab^2 · 1 =a^5
b^11Example B
Simplify the following expression until all exponents are positive.
( 2 x)^5 ·^42
2 −^3 ·a^3 b^2 c^4
a^2 b−^4 c^0Solution:
( 2 x)^5 ·^42
2 −^3 =25 x^524
2 −^3 =29 x^5
2 −^3 =^2(^12) x 5
Example C
Simplify the following expression until all exponents are positive.
(a−^3 b^2 c^4 )−^1
(a^2 b−^4 c^0 )^3
Solution:
(a−^3 b^2 c^4 )−^1
(a^2 b−^4 c^0 )^3 =
a^3 b−^2 c−^4
a^6 b−^12 =
b^10
a^3 c^4
Concept Problem Revisited
Consider the following pattern and decide what the next term in the sequence should be:
16, 8, 4, 2, ___
It makes sense that the next term is 1 because each successive term is half that of the previous term. These numbers
correspond to powers of 2.
24 , 23 , 22 , 21 ,
In this case you could decide that the next term must be 2^0. This is a useful technique for remembering what happens
when a number is raised to the 0 power.
One question that extends this idea is what is the value of 0^0? People have argued about this for centuries. Euler
argued that it should be 1 and many other mathematicians like Cauchy and Möbius argued as well. If you search
today you will still find people discussing what makes sense.
In practice, it is convenient for mathematicians to rely on 0^0 =1.