MAT 1275: Introduction to
Mathematical Analysis
Dr. A. Rozenblyum
II. Exponents and Radicals
A. Using Integers as Exponents
Using Integers as Exponents
Let’s start with the well-known notation of multiplication. Everybody knows that
3  4  12 . But do you remember how we got this 12? Yes, we got it by adding number 3
to itself 4 times: 3 + 3 + 3 + 3 =12. Multiplication means repetition with addition. It
allows us to write an expression in which we need to summarize the same number several
times, in a short and compact form.
There are situations in which we need repetition with multiplication, when we want to
multiply a number by itself several times. For example, consider the product 3  3  3  3 .
It would be a good idea to invent a special notation, similar to multiplication, that allows
to write such a product in a short form, using only number 3 (which tells us the number
that we want to multiply by itself) and number 4 (which tells us how many time to
multiply). We cannot use the notation 3 4 because it already taken for multiplication.
The following notation was invented:34 . Notice that here number 4 is not written next to
number 3, but instead above it and to the right. This expression is called the exponent. It
can be also read as “three to the fourth power”. So, by definition
34  3  3  3  3 .
(In some computer languages and calculators, to keep both numbers on one line, the
notation 3^4 is used).
Number 3, which is on the bottom of the exponent 34 , is call the base, and the number 4,
which is on the top of the exponent, is called the power. We can say that we rise given
base to the given power. And the result of this operation is exponent.
In a similar way we can define exponent in general form, for arbitrary number a and
arbitrary positive integer n:
a n  a  a  ...  a (multiply n times).
Again, here a is the base, and n is the power.
Using Integers as Exponents
The notation of exponents is useful in many situations, in particular, when we work with very big numbers (for
example, with distances between planets). Bellow we will show a way how exponents can be also used for very small
numbers (such as, for example, as distances inside atoms).
Let’s consider examples, and study some properties of exponents.
Example 1. Represent the following numbers in the form of exponents:
a) 100,
b) 1000.
c) 10000.
d) 10 … 0 (n zeros)
Solution.
a) Obvious that 100  10  10 , so 100  102 .
b) Similar, 1000  10  10  10 , so 1000  103 .
c) We have 10000  10  10  10  10 . Therefore, 10000  104 .
d) 10 ...0  10  10  ...  10 (multiply n times). Therefore, 10...0  10n .
Note. Notice, that in all these examples the power is equal to the number of zeros in given number.
Using Integers as Exponents
Example 2. Some people believe that one kilobyte (KB) of computer memory is equal to
1000 bytes (B). However, 1 KB = 1024 B, not 1000 B. The reason is that the number
1024 is a power of 2 but the number 1000 is not. Express number 1024 in exponential
form with the base of 2.
Solution. Let’s divide 1024 by 2 several times: 1024  2  512 , 512  2  256 , and so on.
We will get that 1024 is the product of 10 copies of 2, so 1024  210 .
Example 3. What is the meaning of a1 ?
Solution. By definition, to get a1 we need to repeat number a as a factor one time. It
simply means the same number a. So, for any number a, a1  a .
Note. Compare the above answer with the multiplication by 1: a 1 means to repeat the
number a with addition one time, which also gives a.
Example 4. It is known that the distance from our Earth to the Sun is about 150,000,000 km (150
million kilometers). Represent this number in a short form using exponents.
Solution. We can write this distance in the form 150,000,000  1.5  100,000,000 .
Number 100,000,000 contains 8 zeros. According to example 1d), we have
100 ,000 ,000  10 8 .
8
Therefore, 150 ,000 ,000  1.5  10 .
Note. Representation of big numbers such as in the example 4 in exponential form with the base of 10 is
widely used in science. This form is called scientific notation. We will discuss scientific notation in
more details latter.
Using Integers as Exponents
Now, how about very small numbers? Consider, for example, the diameter of DNA helix. It is
known that this diameter is about 0.0000002 cm. Is it possible somehow to represent this number
also in a short form using exponents? It turns out that the answer is yes. However to come up
with the idea how to do this we need to develop more theory of exponents.
Let’s start with some simple properties.
Basic Properties of Exponents
Working with exponents, we can operate with them like with any other numbers, in particular, make
arithmetic calculation: add them, subtract, multiply, and divide. Often, we want to keep the answers also in
exponential form. As for addition and subtraction, there are no special simple techniques to do that.
However, for multiplication and division, in some cases we can do these operations very easily
Assume that we need to multiply the following two exponents: 34  35 . Is it possible to
write the answer also in the form of exponent? If you recall the definition of exponents,
4
the answer becomes obvious: of course, yes. Indeed, in 3 we have four factors of 3, and
5
4
5
in 3 – 5 factors. So, in 3  3 , we have 4 + 5 = 9 factors of 3. Therefore,
34  35  345  39 .
We see that to write the answer in the exponential form, we just need to sum up powers.
The same rule is true in general case.
Using Integers as Exponents
Multiplication Rule. For any number a, and any positive integers n and m,
a n  a m  a nm .
Note. Notice that all exponents in the above formula have the same base a. This
restriction is very important. If, for example, you need to multiply 34  25 , there is no
simple rule to represent the answer as exponent.
Note. Notice how this multiplication rule works: to multiply exponents with the same
base, we add powers. A common mistake here is to think that we need to multiply
4
5
20
powers. If you believe that “3  3  3 ”, you, probably, have to re-read this text from
the very beginning.
Another simple but useful rule is how to raise exponents into a power.
Power Rule. For any number a and any positive integers n and m,
( a n ) m  a n m .
So, this time, contrary to multiplication rule, we multiply exponents.
Example 5. According to Kepler’s Third Law of planetary motion, the following ratio k
T2
is the same for all planets: k  3 , where T is the orbital period (the time required for a
R
planet to orbit the Sun), and R is the radius of orbit (average distance from a planet to the
7
11
2
3
Sun). For Earth, T  3.16 10 s (seconds), and R  1.5 10 m. Calculate T and R .
Solution. T 2  (3.16 107 ) 2  3.162  (107 ) 2  10 1014  1015 (s 2 ) ,
R 3  (1.5 1011 )3  1.53  (1011 )3  3.36 1033 (m3 ) .
Using Integers as Exponents
Now consider division of exponents. Suppose, we need to divide 36 by 32 . We can
write this operation using division symbol or using fraction:
36
36  32  2 .
3
Similar to multiplication, we see from the definition of exponents, that on the top of the
fraction (in numerator) we have six factors of 3, and on the bottom (in denominator) we
have two factors of 3. After cancellation, we will have 6 – 2 = 4 factors of 3. So,
36
 362  34 .
2
3
The same rule is true in general case. The only restriction is that the power of numerator
should be greater than the power of denominator. And, of course, all exponents must have the
same base.
Division Rule. For any nonzero number a, and any positive integers n and m, such that
n > m,
an
 a nm .
m
a
So, to divide exponents, we subtract powers (we do not divide them).
Note. We set the restriction that a is nonzero, because if a = 0, the above formula does not
make sense (we cannot divide by zero). This is the only restriction on a.
Using Integers as Exponents
As we already noted, in the above formula, the power of numerator must be greater than
the power of denominator. But what if we need to divide exponents in which the power
of numerator is less than the power of denominator? For example, what can we do about
32
?
36
One possible way is just to reduce this fraction, and we will get
32 1

.
36 34
However, this is not what we want: we want to write the answer also in the form of
exponent. If we try to formally use the equation from the Division Rule, and ignore the
restriction n > m, we will have
32
 32  6  3 4 .
36
We come up to the exponent with negative power! You may say that it does not make
any sense. Indeed, by the original definition of exponent, its power tells us how many
times to multiply the base by itself. How can we multiply anything “negative number of
times”?
Of course, we cannot. However, there is a way to give a sense to the above formula with
negative power. This way is based on one of the remarkable idea in mathematics: if we
have a formula that is based on some definition, and we want to generalize this formula
for new objects for which the original definition does not make sense, we can ignore the
original definition, and instead use the formula as a new definition for new objects.
Actually, you already know an example of such approach. Recall the multiplication by
zero. You know that for any number a, a  0  0 . Have you ever thought why is that? Try
to use the original definition of multiplication as repetition of addition. You need to add
the number a to itself “zero number of times”. Of course, it does not make sense.
However, we use the formula a  0  0 all the time. We are doing this just by definition,
and ignore here the original definition of multiplication.
Using Integers as Exponents
We can do similar thing with negative powers. Let’s return to our “strange” formula
32
 3 4 .
6
3
1
We already reduced the left side of this expression to 4 . Therefore,
3
1
1
 3 4 , or 3 4  4 .
4
3
3
Now we can accept this formula as a definition of 34 : it is reciprocal to 34 . The
same reasoning can be used in general form for arbitrary negative powers, and in this way
we invent the following definition.
Definition. For any nonzero number a, and for any positive integer n, we define a  n as
1
a n  n .
a
So, exponents with negative powers are reciprocals to exponents with positive powers.
Now, let’s try to invent the definition of a 0 (exponent with zero power). We can use
similar approach as with negative exponents, using Division Rule for m = n. We will
an
an
0
have n  a n  n  a 0 . Because n  1 , we have a  1 . We come up with the definition.
a
a
0
Definition. For any nonzero number a, a  1 .
Using Integers as Exponents
Example 6. Calculate
a) 10 0
b) 10 1
2
c) .10
n
d).10
Solution. Just by definition, we have
a) 100  1 .
b) 10 1 
1
 0.1 .
10
c) . 10  2 
d). 10 n 
1
1

 0.01 .
2
10
100
1
 0.0...01 (n – 1 zeros after decimal point).
10 n
Example 7. Using example 5, calculate the Kepler’s constant k.
Solution. We have k 
T2
1015
1
1015
1




 101533  0.298  10 18 .
3
33
33
R
3.36  10
3.36 10
3.36
Using Integers as Exponents
Let’s return to the example with the diameter of DNA helix, which is 0.0000002 cm.
Example 8. Represent number 0.0000002 in the form of exponents.
Solution. We can proceed like this
0.0000002 
2
2
1
 7  2  7  2 10 7 .
10000000 10
10
This and previous examples show that exponents with negative powers are useful for
representation small numbers in a compact form. This method, as for big numbers, is
also very common in science. The above representation is also called scientific
notation. Let’s consider this notation in more details.
Using Integers as Exponents
Scientific Notation
As we saw in examples 4, 5, 7 and 8, exponents allow to represent both big and small
numbers in a short and compact form. However, this form is not unique. For example,
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number 5,4000,000 can be written as 54  10 , or 5.4  10 or 0.54  10 and so on. It
would be a good idea to uniformize the exponential representation. In such
representation, given number is written as product of two parts: a number, which is
not very big or very small, times a power of 10. It was decided that the first part
should always be in the range from 1 to 10 (1 is included, but 10 is not). This is
exactly what we mean by scientific notation.
Definition. Number a 10 n (n is any integer) is written in scientific notation, if
number a is in the interval from 1 to 10:
1  a  10 .
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For example, for the number 5,4000,000, the representation 5.4 10 is a scientific
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8
notation, but representations 54 10 and 0.54  10 are not.
Example 9. It is known that the speed of light is about 1,080,000,000 km/h. Express
this number in scientific notation.
Solution. It is clear that in representation a  10 n , a = 1.08. The main problem is to
find the power n. To do this we can just count number of digits in the number
1,080,000,000 between “old” position of decimal point (which is at the end of this
number), and “new” position (which is between digits 1 and 0). We count 9 digits.
The original number is a big number, hence n is positive, so n = 9. Therefore,
1,080 ,000 ,000  1.08  10 9 (km/h).
Using Integers as Exponents
Example 10. It is known that the diameter of carbon atom is about 0.00000000022 m.
Express this number in scientific notation.
Solution. In the representation a  10 n , a = 2.2. To find the power n, as in example 9,
we count number of digits in the original number between “old” position of decimal
point (which you can see at the beginning of the number between first 0 and second
0) and “new” position (which is between 2 and 2). We count 10 digits. The original
number is small number, hence n is negative, so n = – 10. Therefore,
0.00000000022  2.2  10 10 (m).
Example 11. The Kepler’s constant k, calculated in example 7, looks like written in
scientific notation. However this is not true (Why?). Express the constant k in
scientific notation.
Solution. We have k  0.298 10 18 . We need to express the number 0.298 through a
number in the interval from 1 to 10: 0.298  2.98 10 1 . From here,
k  0.298 10 18  2.98 10 1 10 18  2.98 10 118  2.98  10 19 .
End of the Topic