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Exponents and Radicals
Integer Exponents:
n
 Definition 1: a , n is an Integer (Z) and a is a Real number (R)
o For n a positive integer and a a real number:
n
 a  a a ... a (n factors of a)
1

an  n
a
a0

0
a0
  a  1
 Examples:
2 0
oWrite in decimal
form (x )

  1 
5
o Write in decimal form 10
 0.00001
 1 
 4 

o Write using positive
exponents x 

u7

v 3
o Write using positive exponents


Theorem 1: Properties of Integer Exponents
a and b real numbers
o For n and m integers and
m n
mn
 a a a
an 
m









 a nm
(ab) m  ambm
a m a m 

   
m 
b  
b 
 mn
m a
a
  1
a n  nm

a
b0
a0
Examples:
o Simplify eachexpression using positive exponents only:
5 2
  x x


b2 

 
4




a 2b 3

5

a 3
 
b 



m2n 3 2

 4 1 

m n 

Scientific Notation
n

1 a 10, n an integer, a in decimal form
 a  10
 this is used for in the science field when working with large numbers.
 Examples:
o Write the numbers in scientific notation


 58,620,000

 0.000000064

o Write each number in standard form
3
 4  10
 0.004
5
 2.99  10
 299,000

Roots and Real Numbers:
 Definition of an nth Root

o For a natural number n and a and b real numbers
n
 A is the nth root of b if a  b
 Theorem 2: Numbers of Real nth Roots of a Real Number b
o b>0 If n is even, then b has two real roots. If n is odd, then
b has one real root. 
o b=0 0 is the only nth root of 0
o b<0 If n is even, then b has no real nth root; if n is odd, then
b has one real nth root.
 Examples:
2
o a 4
3
o a 8
2
o a  4
 o a3  8
Exponents and Radicals
Rational
 Notation:

1
b2
n
o
o
b
Principal nth Root
 For n a natural number b a real number, the principal nth root of
b is

o The nth root of b if there is only one
o The positive nth root of b if there are two real nth roots
o Undefined if b has no real nth root.
and
, Rational Number Exponent
 For m and n natural numbers and b any real number (except b
cannot be negative when n is even):
o

Example:
(Note:
o
is an exponent in both parts of the
equation.)

Example:
Simplifying Radicals
 Properties of Radicals
o For n a natural number greater than 1, and x and y positive
real numbers





Simplified (radical) Form
o No radicand contains a factor to a power greater than or
equal to the index of the radical.
o No power of the radicand and the index of the radical have a
common factor other than 1.
o No radical appears in a denominator.
o No fraction appears with in a radical.
Rationalizing the denominator
o Eliminating a radical from the denominator.
o To rationalize the denominator, we multiply the numerator
and denominator by a suitable factor that will leave the
denominator free of radicals.
o Rationalizing factor
 The suitable factor that leaves the denominator free of
radicals.
o If the denominator is of the form
, then
is the
rationalizing factor.
o If the denominator is of the form
, then
is the
rationalizing factor.
o CHECK OUT MATHCED PROMBELMS ON PAGE 19