This document defines exponents and radicals. It discusses exponential notation, zero and negative exponents, and the laws of exponents. It also covers scientific notation, nth roots, rational exponents, and rationalizing the denominator. The objectives are to define integer exponents and exponential notation, zero and negative exponents, identify laws of exponents, write numbers using scientific notation, and define nth roots and rational exponents.

1. Exponents and Radicals
Section 1.2

2. Objectives
Define integer exponents and exponential
notation.
Define zero and negative exponents.
Identify laws of exponents.
Write numbers using scientific notation.
Define nth roots and rational exponents.

3. Exponential Notation
an
= a * a * a * a…* a (where there are n
factors)
The number a is the base and n is the
exponent.

4. Zero and Negative Exponents
If a ≠ 0 is any real number and n is a
positive integer, then
 a0
= 1
 a-n
= 1/an

5. Laws of Exponents
am
an
= am+n
 When multiplying two powers of the same base,
add the exponents.
am
/ an
= am – n
 When dividing two powers of the same base,
subtract the exponents.
(am
)n
= amn
 When raising a power to a power, multiply the
exponents.

6. Laws of Exponents
 (ab)n
= an
bn
 When raising a product to a power, raise each factor to
the power.
 (a/b)n
= an
/ bn
 When raising a quotient to a power, raise both the
numerator and denominator to the power.
 (a/b)-n
= (b/a)n
 When raising a quotient to a negative power, take the
reciprocal and change the power to a positive.
 a-m
/ b-n
= bm
/ an
 To simplify a negative exponent, move it to the opposite
position in the fraction. The exponent then becomes
positive.

7. Scientific Notation
Scientific Notation—shorthand way of
writing very large or very small numbers.
 4 x 1013
 4 and 13 zero’s
 1.66 x 10-12
 0.00000000000166

8. nth root
If n is any positive integer, then the
principal nth root of a is defined as:

 If n is even, a and b must be positive.
means nn
a b b a= =

9. Properties of nth roots





if n is odd
| | if n is even
n n n
n
n
n
m n mn
n n
n n
ab a b
a a
b b
a a
a a
a a
=
=
=
=
=

10. Rational Exponents
For any rational exponent m/n in lowest
terms, where m and n are integers and
n>0, we define:

 If n is even, then we require that a ≥ 0.
/ nm n m
a a=

11. Rationalizing the Denominator
We don’t like to have radicals in the
denominator, so we must rationalize to get
rid of it.
Rationalizing the denominator is
multiplying the top and bottom of the
expression by the radical you are trying to
eliminate and then simplifying.

12. Classwork

13. Homework