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Exponents
The mathematician’s shorthand
Is there a simpler way to write
5 + 5 + 5 + 5?
4·5
Just as repeated addition can be simplified by multiplication,
repeated multiplication can be simplified by using
exponents. For example:
2 · 2 · 2 is the same as 2³, since there are three 2’s being
multiplied together.
Likewise, 5 · 5 · 5 · 5 = 54, because there are four 5’s
being multiplied together.
Power – a number produced by raising a base to
an exponent. (the term 27 is called a power.)
Exponential form – a number written with a base
and an exponent. (23)
Exponent – the number that indicates how many
times the base is used as a factor. (27)
Base – when a number is being raised to a power,
the number being used as a factor. (27)
Evaluating exponents is the second step in the
order of operations. The sign rules for
multiplication still apply.
Writing exponents
3 · 3 · 3 · 3 · 3 · 3 = 36
(-2)(-2)(-2)(-2) = (-2)4
x · x · x · x · x = x5
12 = 121
How many times is 3 used as a factor?
How many times is -2 used as a factor?
How many times is x used as a factor?
How many times is 12 used as a factor?
36 is read as “3 to the 6th power.”
Evaluating Powers
26 = 2 · 2 · 2 · 2 · 2 · 2 = 64
83 = 8 · 8 · 8 = 512
54 = 5 · 5 · 5 · 5 = 625
Always use parentheses to raise a negative number
to a power.
(-8)2 = (-8)(-8) = 64
(-5)3 = (-5)(-5)(-5) = -125
(-3)5 = (-3)(-3)(-3)(-3)(-3) = -243
When we multiply negative numbers together,
we must use parentheses to switch to
exponent notation.
(-3)(-3)(-3)(-3)(-3)(-3) = (-3)6 = 729
You must be careful with negative signs!
(-3)6 and -36 mean something entirely different.
Note:
When dealing with negative numbers,
*if the exponent is an even number the answer
will be positive.
(-3)(-3)(-3)(-3) = (-3)4 = 81
*if the exponent is an odd number the answer
will be negative.
(-3)(-3)(-3)(-3)(-3) = (-3)5 = -243
In general, the format for using
exponents is:
(base)exponent
where the exponent tells you how
many times the base is being
multiplied together.
Just a note about zero exponents: powers such as 20, 80
are all equal to 1. You will learn more about zero powers
in properties of exponents and algebra.
Simplifying Expressions Containing
Powers
• Simplify 50 – 2(3 · 23)
50 – 2(3 · 23)
= 50 – 2(3 · 8) Evaluate the exponent.
= 50 – 2(24) Multiply inside parentheses.
= 50 – 48
Multiply from left to right.
=2
Subtract from left to right.
Simplify and Solve
1) (3 - 62) =
2) 42 + (3 · 42)
3) 27 + (2 · 52)
4) (-3)5
5) 2(53 + 102)
• A population of
bacteria doubles in
size every minute.
The number of
bacteria after 5
minutes is 15(25). How
many bacteria are
there after 5
minutes?
Properties of
Exponents
Multiplying, dividing powers and
zero power.
The factors of a power, such as 74, can be
grouped in different ways. Notice the
relationship of the exponents in each product.
7 · 7 · 7 · 7 = 74
(7 · 7 · 7) · 7 = 73 · 71 = 74
(7 · 7) · (7 · 7) = 72 · 72 = 74
Multiplying Powers with the Same Base
• To multiply powers with the same base, keep the
base and add the exponents.
• 35 · 38 = 35+8 = 313
• am · an = a m+n
Multiply
• 35 · 32 = 35+2 = 37
• a10 · a10 = a10+10 = a20
• 16 · 167 = 161+7 = 168
• 64 · 44 =
Cannot combine; the bases are not the same.
Dividing Powers with the Same Base
• To divide powers with the same base, keep the
base and subtract the exponents.
• 69 = 69-4 = 65
64
• bm = bm-n
bn
Divide
• 1009 = 1009-3 = 1006
1003
• x8 =
y5
Cannot combine; the bases are not the same.
When the numerator and denominator of a fraction
have the same base and exponent, subtracting the
exponents results in a 0 exponent.
1 = 42 = 42-2 = 40 = 1
42
•The zero power of any number except 0
equals 1.
1000 = 1
(-7)0 = 1
a0 = 1 if a ≠ 0
How much is a googol?
10100
Life comes at you fast, doesn’t it?
Negative Exponents
Extremely small numbers
Negative exponents have a special meaning.
The rule is as follows:
Basenegative exponent =
Base1/positive exponent
4-1 = 1
41
Look for a pattern in the table below to
extend what you know about exponents. Start
with what you know about positive and zero
exponents.
103 = 10 · 10 · 10 = 1000
102 = 10 · 10 = 100
101 = 10 = 10
100 = 1 = 1
10-1 = 1/10
10-2 = 1/10 · 10 = 1/100
10-3 = 1/10 · 10 · 10 = 1/1000
Example:
10-5 = 1/105 = 1/10·10·10·10·10 = 1/100,000 = 0.00001
So how long is 10-5 meters?
10-5 = 1/100,000 = “one hundred-thousandth of a meter.
Negative exponent – a power with a negative exponent
equals 1 ÷ that power with a positive exponent.
5-3 = 1/53 = 1/5·5·5 = 1/125
Evaluating negative exponents
1)
(-2)-3 = 1/(-2)3 = 1/(-2)(-2)(-2) = -1/8
2)
5-3 = 1/53 = 1/(5)(5)(5) = 1/ 125
3)
(-10)-3 = 1/(-10)3 = 1/(-10)(-10)(-10) = -1/1000 =
0.0001
4)
3-4 · 35 = 3-4+5 = 31 = 3 Remember Properties of Exponents: multiply same
base you keep the base and add the exponents.
Evaluate exponents:
Get your pencil and calculator ready to solve
these expressions.
1)
2)
3)
4)
5)
6)
7)
8)
10-5 =
105 =
(-6)-2 =
124/126 =
12-3 · 126
x9/x2 =
(-2)-1 =
23/25 =
Problem Solving using exponents
The weight of 107 dust particles is 1 gram. How
many dust particles are in 1 gram?
As of 2001, only 106 rural homes in the US had
broadband internet access. How many homes had
broadband internet access?
Atomic clocks measure time in microseconds. A
microsecond is 0.000001 second. Write this
number using a power of 10.
Exponents can be very
useful for evaluating
expressions, especially
if you learn how to use
your calculator to work
with them.