Integer Exponents and Rules of Exponents

I.  Quiz Return

II.  Homework

III.  Integer Exponents

Definition:  x^-1 = 1/x, and x^-n 1/xⁿ

Example:  3^-1 = 1/3, 2^-3 = 1/2³ = 1/8

Definition:  x⁰ = 1 for all nonzero x.

III.  Exponent  Rule 1

xⁿx^m = x^{n+ m} 

Examples:  2³2⁵ = 2^{3 + 5}   = 2⁸

w²w³ = w⁵

xy² x³y³x⁴y⁴ = x⁸y⁹

III.  Rule 2

(xⁿ)^m =   x^nm     

Examples:

A)  (2⁴)² = 2⁸

B)  (x²)³  = x⁶ 

C)  4² 2³ 8 = (2²)² 2³ 2³ = 2⁴ 2³ 2³ = 2¹⁰    

IV.  Rule III

(xy)ⁿ = xⁿyⁿ    

Examples:

A)    (6)⁴ = (2)⁴(3)⁴

B)  (3x²)³ = 3³ (x²)³  = 27x⁶         

C)  Try  (-x³)⁴

V.  Rule IV

For x not 0 we have

xⁿ/x^m = x^n-m

Examples:

x³/x² = x^3-2  = x¹ = x

x⁴/x⁸ =  x^{4 - 8} =  x^-4 = 1/x⁴

Application:  The population of the world can be modelled by the equation

P = A 2^t/30

where t is the time in years after 1970.  If there were 3 billion people in 1970, What will the population be in 2060?

Solution:  We first know that if t is 0 then P is 3 (in billions).  Hence

3 = A 2^0/30 = A

So that

P = 3 (2^t/30)

1060 corresponds to t = 90 so that

P = 3 2^90/30  = 3 (2³) = 3(8) = 24 billion people.