Download Integer Exponents and Scientific Notation Section 0.2

Integer Exponents and
Scientific Notation
Section 0.2
What’s an exponent?
Exponents are shorthand notation for repeated
multiplication:
5555 = 54
There are four 5’s being multiplied together.
In 54 , the 5 is called the base and the 4 is the
power or exponent.
In 5555 , the 5’s are called factors.

Evaluating expressions
Evaluating an expression means to find out
what it’s worth (giving it’s value)…just do
the math.
Evaluate the following:
4 2  4  4  16
 4 2  (4  4)  16
 42
(note that the location of
the negative sign and the
parenthesis make a
difference in the answer!)
 (4)  (4)  16

Evaluating expressions continued
Evaluate the following:
3234
This can become:
or:
Which is:
333333
36
729
This idea is called the Product Property of
Exponents. When you are multiplying exponentials
with the same base you add the exponents. Just
remember the bases MUST be the same.
a a  a
m
n
m n

More properties of exponents
6
5
4
5
This can become:
am
mn
a
n
a
555555
2
5
5555
Remember that a number
divide by itself is 1…
So all that is left is 55 which is 25. This is the
Quotient Properties of Exponents. When you
divide exponentials with the same base,
subtract the exponents.

More properties of exponents
Power property of exponents:

More properties of exponents

EXAMPLES
a.
(–4
Evaluate numeric expressions
25)2 = (– 4)2 (25)2
 16  210
 16384
b.
115
118
–1
115
 8
11
118
 5
11
= 113 = 1331

Simplifying Algebraic Expressions
Algebraic expressions are simplified when
the following things have happened or are
“done”:
All parenthesis or grouping symbols have
been eliminated
A base only appears once
No powers are raised to other powers
All exponents are positive

EXAMPLES
a.
b.
c.
Simplify algebraic expressions
b–4b6b7
r–2
s3
= b–4 + 6 + 7 = b9
–3
Product of powers property
–2 )–3
(
r
=
( s3 )–3
Power of a quotient property
6
r
= –9
s
Power of a power property
= r6 s9
Negative exponent property
16m4n –5
4n –5 – (–5)
=
8m
2n–5
= 8m4n0= 8m4
Quotient of powers property
Zero exponent property

EXAMPLE
Standardized Test Practice
SOLUTION
(x–3y3)2
x5 y 6
–6y6
x
=
x5 y6
The correct
answer is B
 x 11 y 0
1
 11
x

More Examples
x y 
x y 
3
2 2
4
3 3
x 6 y 4
x 12 y 9
x 6  x12  y 9
y4
x18 y 5
More Examples
 4 x  y z
2 x  y z 
2 2
3 2
3
1
4
 42 x 4 y 3 z
2
6
4 4
2 x y z
16  2 2 x 6 y 3 y 4 z
x4 z 4
64 x 2 y 7 z 3
1
64 x 2 y 7
z3
GUIDED PRACTICE
Simplify the expression. Tell which properties of
exponents you used.
x–6x5 x3
ANSWER x2 ; Product of powers property
(7y2z5)(y–4z–1)
ANSWER
7z4 ; Product of powers property,
y2 Negative exponent property

GUIDED PRACTICE
s3
2
t–4
ANSWER
x4y–2
x3 y6
s6t8 ; Power of a power property,
Negative exponent property
3
ANSWER
x3
y24
; Quotient of powers property,
Power of a Quotient property,
Negative exponent property

Scientific Notation
Scientific Notation was developed in order to
easily represent numbers that are either very
large or very small. Following are two
examples of large and small numbers. They
are expressed in decimal form instead of
scientific notation to help illustrate the problem

A very large number:
The Andromeda Galaxy (the closest one to our Milky Way galaxy)
contains at least 200,000,000,000 stars.
A very small number:
On the other hand, the weight of an alpha particle, which is
emitted in the radioactive decay of Plutonium-239, is
0.000,000,000,000,000,000,000,000,006,645 kilograms.
As you can see, it could get tedious writing out those numbers
repeatedly. So, a system was developed to help represent these
numbers in a way that was easy to read and understand:

Scientific Notation.
Decimal to Scientific Notation
Move the decimal point so the number
shown is between 1 and 10
 Count the number of spaces moved and
this is the exponent on the 10
 If the original number is bigger than 1, the
exponent is positive
 If the original number is between 0 and 1,
then the exponent is negative.

What to do for scientific notation
Write in scientific notation: 200,000,000,000
So we write the number in scientific notation as
2.0 x 1011
Write in scientific notation:
0.000,000,000,000,000,000,000,000,006,645
6.645 x 10-27

Scientific Notation to Decimal
The number of spaces moved is the
exponent on the 10
 Move to the right if the exponent is
positive
 Move to the left if the exponent is
negative

6.45 x 104 = 64,500
2.389 x 10-6
= .000002389