3 Exponents - Revised

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MAT 85 – Introduction to Mathematics for Computer Science

Lecture #3 – Exponents

MAT85 – Introduction to Mathematics for Computer Science Lecture 3


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



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



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!



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



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?



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



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



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



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



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

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3 Exponents - Revised

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