Properties of Exponents

1
Properties of Exponents

• Let a and b be real numbers and m and n be
  rational
• numbers
• 1. am an amn 2. (am)n
  amn 3. (a b)m am bm
  4. am / an amn 5. ( a/b )m am
  / bm 6. am 1/am
• 7. a0 1
• Denominators cannot be equal to zero
• See Appendix F on page 1021 of your text.

2
Square Roots
The principal square root of a nonnegative number
is its nonnegative square root. The symbol va
represents the principal square root of a. The
negative square root of a is - v a
3
Square Roots
4
Examples of Square Roots

• v(-16)² -16 16
• v (3b)² 3b 3b
• v 16y² 4y 2y
• v(x-1)² x-1
• v(x7)² x7
• v x²8x16 x4
• v x²6x9 x3

5
nth Roots

• 3 is the square root of 9 because 9 is 3
  squared
• Roots exist other than square roots. For
  example
• 2 is the cube root of 8 since 23 8
• 5 is a fourth root of 625 since 54 625
• 5 is a fourth root of 625 since (5)4 625

6
nth Roots
7
nth Roots

• There is only one real number cube root for each
  real number
• When the index is even (square root, 4th root,
  and so on), the radicand must be nonnegative to
  yield a real number root.

8
nth Roots and Rational Exponents

• 3 is the square root of 9 because 9 is 3
  squared
• Roots exist other than square roots. For
  example
• 2 is the cube root of 8 since 23 8
• 5 is a fourth root of 625 since 54 625
• 5 is a fourth root of 625 since (5)4 625
• These roots can be written with two different
  types of notation
• Radical Notation or
  Rational Exponent Notation

9
Rational Exponents
For any exponent of the radicand, the rational
exponent form of a radical looks like this
If m and n are positive integers with m / n in
lowest terms then

If all indicated roots are real numbers then
10
To simplify rational exponents, you may use the
following If m and n are positive integers
with m/n in lowest terms, then am/n (a1/n)m
Rational Exponents

• Example Simplify 82/3
• (81/3)2
• (2)2
• 4
• Example Simplify 64-2/3
• (641/3)-2
• (4)-2
• 1 (4)2
• 1 16

11
Rational Exponents
The basic properties for integer exponents also
hold for rational exponents as long as the
expression represents a real number.
See Appendix F on page 1021 of your text.
12
Examples

• Simplify 641/3
• This is the same as (43)1/3 4
• or
• 641/3

4
Simplify 625 1/4 This is the same as
1 625 1/4 or
1 5
13
Rational Exponents
How do you simplify ?

• Reduce the rational exponent, if possible.

• You can rewrite the expression using a radical.

• Simplify the radical expression, if possible.

14
Examples
No real number solution
15
Rational Exponents
Example
What would the answer above be if you were to
write it in radical form?