Laws of Exponents
Objective:
TSW simplify powers.
TSW simplify radicals.
TSW develop a vocabulary associated
with exponents.
TSW use the laws of exponents to
simplify.
Exponents

The lower number is called the base and the
upper number is called the exponent.

The exponent tells how many times to
multiply the base.
Exponents
exponent
7
3





1. Evaluate the following exponential
expressions:
A. 42 = 4 x 4 = 16
B. 34 = 3 x 3 x 3 x 3 = 81
C. 23 =
7
D. (-1) =
Squares

To square a number, just multiply it by itself.
=
3 squared =
= 3x3=9
Perfect Squares







1² = 1
2² = 4
3² = 9
4² = 16
5² = 25
6² = 36
7² = 49






8² = 64
9² = 81
10² = 100
11² = 121
12² = 144
13² = 169
Square Roots

A square root goes the other direction.

3 squared is 9, so the square root of 9 is 3
3
9
Square Roots
1 1
4 2
9 3
16  4
25  5
36  6
49  7
64  8
81  9
100  10
121 11
144  12
169  13
Radicals
- The inverse operation of raising a number to a
power.

For Example, if we use 2 as a factor with a power of
4, then we get 16. We can reverse this by finding
the fourth root of 16 which is 2.
4
16
= 2
Radicals



In this problem, the 16 is called the radicand,
the 4 is the index, and the 2 is the root.
The symbol is known as the radical sign. If
the index is not written, then it is understood
to be 2.
The entire expression is known as a radical
expression or just a radical.
Example

a)
Simplify:
c)
4
3
81
b)
16
d)
27
3
8
Laws of Exponents

Whenever we have variables which contain
exponents and have equal bases, we can do
certain mathematical operations to them.

Those operations are called the “Laws of
Exponents.”
Laws of Exponents
1. x  x  x
n
m
3. x

m n
m n
2. xy   x y
m
m
x
mn
m
x
x
4.    m
y
 y
m
m
Laws of Exponents
m
x
mn
5a. if m  n , then n  x
x
m
x
1
5b. if n  m , then n  n  m
x
x
Zero Exponents
a

0
= 1
A nonzero based raise to a
zero exponent is equal to one
Negative Exponents
a
-n
=
(
1
______
n
a
)
A nonzero base raised to a negative
exponent is the reciprocal of the base
raised to the positive exponent.
Basic Examples
x x  x
2
3
x 
4 3
x
23
43
x
x
5
12
Basic Examples
xy
3
3
x y
3
x
x
   3
y
 y
3
3
Basic Examples
7
74
x
x
3


x
4
x
1
5
x
1
1


7
7 5
2
x
x
x
Examples
5
1.
u
  
 p
2.
y 
3.
x x 
7 5
4
9
4.

7
x

3
x
Scientific Notation
Objective:
TSW rewrite numbers in scientific notation
TSW perform operations with numbers in
scientific notation.
TSW solve real-world problems using
numbers in scientific notation.

When we multiply a number by a positive
power of 10, we move the decimal point to
the right the number of places indicated by
the exponent.

This method of numbers is known as
scientific notation.

When we write a number greater than or
equal to ten in scientific notation, we use
three steps:
 1. place the decimal point just right of the
first nonzero digit
 2. count the number of places the decimal
point moved to the left
 3. multiply the number in step one by 10ª (a
is the number of places the decimal point
moved) to indicate where the decimal point
should be.
Example

Write 7,024,000 in scientific notation
Example

Write 476.23 in scientific notation.

We can also write very small numbers in
scientific notation.

For these, we use negative exponents.

We use 10 with a negative exponent to show
that the decimal point should be moved to the
left.

When we write a number between zero and
one in scientific notation, we use three steps:



1. place the decimal point just to the right of the
first nonzero digit
2. count the number of places the decimal point
moved to the right
3. multiply the number in step one by 10ˉª (a is
the number of places the decimal point moved) to
indicate where the decimal point should be
Example
Write 0.0652 in scientific notation.
Example
1. Write these numbers in standard notation:
a.) 4.6 x 10ˉ³
6
b.) 4.6 x 10
2. Saturn is about 875,000,000 miles from the
sun. What is this distance in scientific
notation?
Answers
1. a.) 0.0046
b.) 4600000
8
2. 8.75 x 10
Computing with Scientific
Notation

You can multiply and divide numbers written
in scientific notation. (Use the Laws of
Exponents!)


To multiply powers with the same bases, add the
exponents
To divide powers with the same base, subtract the
exponents
(3.2 x 10²) x (2 x 10³)

Step 1: Multiply the first pair of factors from
each
(3.2 x 2) = 6.4

Step 2: Multiply the second pair of factors (
the ones written in exponential form)
10² x 10³ = 10

2+3
= 10
5
Step 3: Combine the products
5
6.4 x 10
Examples

5
1. (5.4 x 10 ) x (4.6 x 10³)
-4

2. (8.4 x 10³) x (2.1 x 10 )

3. (1.2 x 10-4 ) x (9.6 x 10²)
Answers



1. 2.484 x 10
2. 4 x 10
7
-7
3. 1.25 x 10
9
Adding and Subtracting

You can also add and subtract with numbers
written in scientific notation as long as the
second factors are the same.
Example

8
About 8.73 x 10 people in the world8 speak
Mandarin Chinese. About 3.22 x 10 people
speak Spanish. In scientific notation, how
many more people speak Mandarin Chinese
than Spanish?
Answer



8
8
(8.73 x 10 ) – (3.22 x 10 )
(8.73 – 3.22) x 10
8
8
5.51 x 10 more people speak Mandarin
Chinese than Spanish
Example

7
The Atlantic Ocean has an area of 3.342 x 10
square miles. The Artic Ocean has an area
6
of 5.105 x 10 square miles. In scientific
notation, what is the combined area of the
two oceans?