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© 2014, DigiPen Institute of Technology. All Rights Reserved. 1
MAT 85 – Introduction to Mathematics for Computer Science
Lecture #3 – Exponents
MAT85 – Introduction to Mathematics for Computer Science Lecture 3
© 2014, DigiPen Institute of Technology. All Rights Reserved. 2
Exponential Notation A common language is needed in order to communicate mathematical ideas clearly and
efficiently. Exponential notation is one example. It was developed to write repeated
multiplication more efficiently. For example, growth occurs in living organisms by the division
of cells. One type of cell divides 2 times in an hour. So in 12 hours, the cell will divide
2 ∗ 2 ∗ 2 ∗ 2 ∗ 2 ∗ 2 ∗ 2 ∗ 2 ∗ 2 ∗ 2 ∗ 2 ∗ 2 times. This can be written more efficiently as 212 .
The notation or way to represent an exponential in the way we recognize today was first
introduced by the French mathematician René Descartes (1596 – 1650) and looks like the
following:
𝑎𝑛
Laws of Exponents The law of exponents states that a number 𝑎 raised to the power 𝑛 is the product of 𝑛 factors,
each equal to the number.
𝑎𝑛 = 𝑎 ∗ 𝑎 ∗ 𝑎 ∗ 𝑎 …………𝑎
𝑎𝑛 is called the power, 𝑛 is the exponent and 𝑎 is the base.
𝑎𝑛
Notes that this is only whenever 𝑛 is a positive integer, if the number is negative, we
would divide.
Example:
7 ∗ 7 ∗ 7 ∗ 7 = 74
As you can see using exponent notation is quite beneficial as it reduces the mathematical
expressions involving products of equal factors and makes calculations easier.
n factors
Power Exponent
Base
4 times
MAT85 – Introduction to Mathematics for Computer Science Lecture 3
© 2014, DigiPen Institute of Technology. All Rights Reserved. 3
Usage Exponential notation is needed to express formulas in all scientific branches. For example, the
volume of a sphere in mathematics is: 4
3𝜋𝑅3
And in physics, energy is represented as:
𝐸 = 𝑚𝑐2
where 𝑚 is the mass and 𝑐 the celerity, commonly as the speed of light.
In addition, in accounting you can find the accumulated amount of money earned in a number, 𝑛,
years:
𝐴 = 𝑃 1 + 𝑖 𝑛
where 𝑖 is the interest and 𝑃 the principal amount.
Operations on exponents In this section the exponents are considered to be non-null positive integers, that is nN
*.
However, in section 3, zero, negative, and fractional exponents will be introduced and examined.
Power of product To find a power of a product, find the power of each factor and then multiply:
𝑎 ∗ 𝑏 ∗ 𝑐 𝑛 = 𝑎𝑛 ∗ 𝑏𝑛 ∗ 𝑐𝑛
Proof:
To show that this is the truth, let’s first consider what 𝑎 ∗ 𝑏 ∗ 𝑐 raised to the power n would
give us:
𝑎 ∗ 𝑏 ∗ 𝑐 𝑛 = 𝑎 ∗ 𝑏 ∗ 𝑐 ∗ 𝑎 ∗ 𝑏 ∗ 𝑐 ∗ 𝑎 ∗ 𝑏 ∗ 𝑐 ∗ ………∗ 𝑎 ∗ 𝑏 ∗ 𝑐
By grouping the identical factors together we get:
(a.b.c)n = (a.a.a.a……..a).(b.b.b.b.b………b).(c.c.c.c.c………c) = a
n.b
n.c
n
With this knowledge in place, the above rule can be extended to any number of factors.
n factors
n factors n factors n factors
MAT85 – Introduction to Mathematics for Computer Science Lecture 3
© 2014, DigiPen Institute of Technology. All Rights Reserved. 4
Example:
𝑝 = 2 ∗ 3 ∗ 6 ∗ 4 3 = 23 ∗ 33 ∗ 63 ∗ 43 = 2985984 Just to verify:
𝑝 = 144 3 = 2985984
Power of a ratio Our next task will be to find the power of a ratio:
𝑎
𝑏 𝑛
To find the power of a ratio, our objective is to express a ratio (a/b) raised to the power n as a
ratio of two powers having respectively a and b as bases and n as the exponent.
Condition:
b 0
𝑎
𝑏 𝑛
=𝑎𝑛
𝑏𝑛
Proof:
To show that this is the truth, let’s first consider what 𝑎
𝑏 raised to the power n would give us:
𝑎
𝑏 𝑛
= 𝑎
𝑏 ∗
𝑎
𝑏 ∗
𝑎
𝑏 ………∗
𝑎
𝑏
= 𝑎 ∗ 𝑎 ∗ 𝑎 ∗ ………∗ 𝑎
𝑏 ∗ 𝑏 ∗ 𝑏 ∗ ………∗ 𝑏
=𝑎𝑛
𝑏𝑛
Example:
3
2
5
=35
25=
243
32= 7.59375
Just to verify:
3
2
5
= 1.5 ∗ 1.5 ∗ 1.5 ∗ 1.5 ∗ 1.5 = 7.59375
The equation holds!
MAT85 – Introduction to Mathematics for Computer Science Lecture 3
© 2014, DigiPen Institute of Technology. All Rights Reserved. 5
Multiplication of powers (same base) The objective for multiplication of powers is to express a product of several powers of same base
with different exponents as a power having the same base and an exponent equal to the sum of
exponents of all the powers:
𝑎𝑚 ∗ 𝑎𝑛 = 𝑎𝑚+𝑛
Proof
Consider the following case:
𝑎𝑚 ∗ 𝑎𝑛
𝑎𝑚 ∗ 𝑎𝑛 = 𝑎 ∗ 𝑎 ∗ 𝑎 ∗ … ∗ 𝑎 ∗ 𝑎 ∗ 𝑎 ∗ ……∗ 𝑎 = 𝑎𝑚+𝑛
With this knowledge in place, the above rule can be extended to any number of factors.
Example:
23 ∗ 25 ∗ 27 = 23+5+7 = 215
However, I do want to remark that unlike the product of powers, the sum of powers does not
lead to any law or property.
Power of a power Following the same line of logic, let’s find the power of a power:
𝑎𝑚 𝑛
Which is to say a number with an exponent raised to another exponent. This may look difficult,
but with the power of a power rule it’s not that bad at all:
𝑎𝑚 𝑛 = 𝑎𝑚∗𝑛
This rule means that you multiply the exponents together and keep the base unchanged.
Proof:
𝑎𝑚 𝑛 = 𝑎𝑚 ∗ 𝑎𝑚 ∗ 𝑎𝑚 ∗ … ∗ 𝑎𝑚
n factors
m + n factors
n factors
m factors
MAT85 – Introduction to Mathematics for Computer Science Lecture 3
© 2014, DigiPen Institute of Technology. All Rights Reserved. 6
By using the property of the multiplication of powers having same base
𝑎𝑚 ∗ 𝑎𝑚 ∗ 𝑎𝑚 ∗ … ∗ 𝑎𝑚 = 𝑎𝑚+𝑚+𝑚+⋯+𝑚 = 𝑎𝑚∗𝑛
Example:
42 3 = 42∗3 = 46 = 4096
Just to verify:
42 3 = 16 3 = 4096
Extending the exponent laws Now that we have an understanding of the fundamental ways of working with exponents with
positive numbers, let’s take some time to look at the other types of exponents we haven’t
covered yet:
Zero
Negative
Fractional
Laws given in Section 2 also apply for these types of exponents as well.
Zero exponent a 0 Let’s start off with when we have a power that is 0:
Condition:
a 0
𝑎0 = 1
Proof:
To prove this, we first need to consider the case:
𝑎0 ∗ 𝑎𝑛 = 𝑎0+𝑛 = 𝑎𝑛
⇒ 𝑎0 leaves 𝑎𝑛unchanged by multiplication
hence, 𝑎0 = 1
Examples:
50 = 1
45−5 = 1
Negative exponent Now, what if the power we are given is negative?
MAT85 – Introduction to Mathematics for Computer Science Lecture 3
© 2014, DigiPen Institute of Technology. All Rights Reserved. 7
𝑎−𝑛
Proof :
Using our previous work of multiplication of powers we can see that:
𝑎𝑛−𝑛 = 𝑎0 = 𝑎𝑛 ∗ 𝑎−𝑛 = 1
So with that in mind, 𝑎−𝑛 is the reciprocal of 𝑎𝑛 :
𝑎−𝑛 =1
𝑎𝑛
The really great benefit that we get from learning this is that it makes the computations involving
ratios of powers easier.
Examples:
25
5−4= 25 ∗ 54
4−3 =1
43
In addition to the above, all of the prefixes for powers of 10 can be considered examples as well:
Power of ten Prefix Abbreviation
Negative power of ten 10-2
10-3
10-6
10-9
10-12
10-15
10-18
10-21
10-24
centi-
milli-
micro-
nano-
pico-
femto-
atto-
zepto-
yocto-
c
m
n
p
f
a
z
y
Positive power of ten 102
103
hecto-
kilo-
H
k
MAT85 – Introduction to Mathematics for Computer Science Lecture 3
© 2014, DigiPen Institute of Technology. All Rights Reserved. 8
106
109
1012
1015
1018
1021
1024
mega-
giga-
tera-
peta-
hexa-
zetta-
yotta-
M
G
T
P
E
Z
Y
Quotient of two powers (same base, different exponents) We already learned from the product of powers property that if we multiply two powers with
the same base we add the exponents together, but what if we divide two powers with the same
base?
𝑎𝑚
𝑎𝑛
Condition:
a 0, m and n are natural integers.
Discussion:
o m > n n
m
a
a = a
m – n positive exponent.
Example: (27/2
4) = 2
7 – 4 = 2
3
o m < n n
m
a
a = a
m – n negative exponent.
Example: (53/5
5) = (5
3 – 5) = (5
-2)
o m = n n
m
a
a = a
m – n = a
0 = 1
Sign of the power an If a > 0 a
n > 0 whatever is the value of n.
If a < 0 :
o n even an > 0
o n odd an < 0
MAT85 – Introduction to Mathematics for Computer Science Lecture 3
© 2014, DigiPen Institute of Technology. All Rights Reserved. 9
Examples:
23 = 8 > 0
−3 3 = −27 < 0
−3 4 = 81 > 0
Fractional Exponents Finally, what if our exponent is not a whole number at all? This extension to the exponent laws
was introduced in 1655 by John Wallis:
𝑎𝑛𝑚
The objective is to express the 𝑚th root of a number ‘𝑎’ to the power 𝑛 as a power having 𝑎 as
the base and 𝑚
𝑛 as the exponent.
𝑎𝑛𝑚 = 𝑎(
1𝑚∗𝑛) = 𝑎𝑛𝑚
Now, the benefit to learning this is that we can substitute expressions involving radicals by the
exponent notation, which makes calculations easier using the laws of exponent.
Conditions:
𝑚 ∈ ℕ
𝑛 ∈ ℤ
The symbol ℕ stands for the set of natural numbers, which is to say a positive whole number.
The symbol ℤ stands for the set of integers, which is to say a whole number (positive or
negative) with no fractions.
If m is even a n must be 0. a
n 0
If a 0 n can be either odd or even.
If a 0 n must be even.
If m is odd no conditions.
Demonstration:
We consider the 𝑚th root of 𝑎𝑛 denoted by 𝑎𝑛𝑚
. By raising it to the power of m we get:
𝑎𝑛𝑚 𝑚
= 𝑎𝑛
𝑎𝑛𝑚 𝑚
= 𝑎𝑛 and 𝑎𝑛 = 𝑛
𝑎𝑚 𝑚
by comparing the two values of a n we obtain:
𝑎𝑛𝑚 = 𝑎𝑛𝑚
MAT85 – Introduction to Mathematics for Computer Science Lecture 3
© 2014, DigiPen Institute of Technology. All Rights Reserved. 10
Examples:
912 = 9 = 3
435 = 435
The number of birds triples each 11 years. If the current number of birds is 300, after n years it
will be:
300 ∗ 3𝑛
11
MAT85 – Introduction to Mathematics for Computer Science Lecture 3
© 2014, DigiPen Institute of Technology. All Rights Reserved. 11
Summary chart Rules Conditions
Integer exponents
am
. an = a
n + m
(am
) n
= a m . n
nm
n
m
aa
a a 0
am
.bm
= (a.b)m
a1 = a
a0 = 1 a 0
n
n
a
1a
a 0
Fractional exponents
a x. a
y = a
x + y a > 0
(a x)
y = a
x.y a > 0
a x. b
x =(a.b)
x a > 0 and b > 0
yx
y
x
aa
a
a > 0