Exponents and Rationals
Take a moment and consider the following…
 What is the square root of 16?
 What is the square root of 9?
 What is the square root of
9
? Can you explain your result?
16
Write your thoughts in your notebook or discuss them with a neighbor.
What are our thoughts for these questions?
Definition:
 Exponent
o A number that tells you how many of certain number to
multiply together
o Ex. 32 = 3 x 3 = 9
How does this apply to fractions?
 When a fraction has an exponent applied to it, it affects both the
numerator and denominator.
3 2
32
4
42
o Ex. ( ) =
=
9
16
Definitions:
 Root
o An action that undoes an exponent
o Ex. √9 = √32 = 3
 Perfect Square
o A number that has a whole number square root
o Ex. 9 since √9 = 3
There are some perfect squares which you will likely know by heart at
this point:
 √100 =
 √16 =
 √4 =
What do you do if you don’t or are unsure if the number is a perfect
square?
Definition:
 Prime Number
o A number with only two factors, 1 and itself
 Factors
o Whole numbers you multiply to produce another number
 Prime Factorization
o A method of breaking a number down into prime numbers
that can be multiplied to produce it
Let’s take a moment and try to brain storm as many prime numbers as
we can.
Consider the following…
 What are the factors for the number 12?
 Do any of these numbers also have factors? What are they?
For the number 12, it could be broken down into the following factors
 1 x 12
 2x6
 3x4
In particular I’m going to focus on the 2 x 6 and 3 x 4. For 2 x 6, can
either number be broken down into factors?
 2 I can’t because it is a prime number
 6 I can  2 x 3 and both of these are prime
 So 2 x 6 really is the same as 2 x 2 x 3
I can do the same thing with the 3 x 4 and I’ll find that it turns into
3 x 2 x 2.
 This is the same set of prime numbers!
This is the process we call prime factorization. We break down a
number into factors until we only have primes left. It’s like a fingerprint
for a number since no two numbers have the same set of prime factors.
Example:
 What are the prime factors of 180?
o One way to tackle prime factorization
is to start with multiplications you
know.
o I know 18 x 10 = 180 so all I’d need to
do is the prime factorization of 18 and
10.
o I’m using something called a factor tree
to show my work this time.
o This means that for 180 we have prime factors of 3, 2, 3, 5
and 2 so
180 = 2 x 2 x 3 x 3 x 5
Let’s try a few together:
 What are the factors of 144? (1-4, 6, 8, 9, 12, 16, 18, 24, 36, 48,
72, 144)
 What are the prime factors of 144? (2 x 2 x 2 x 2 x 3 x 3)
 Write 540 as a product of its prime factors. Use a factor tree to
show the breakdown of factors. (540 = 2 x 2 x 3 x 3 x 3 x 5)
What does this have to do with perfect squares?
 When you encounter a perfect square, you will discover that you
can separate the prime factors into two equal groups.
o Ex. 144
 We just figured out that its prime factors are
2x2x2x2x3x3
 I could split these up into two equal groups made up of
two 2’s and one 3
2 x 2 x 3 and 2 x 2 x 3
 If I figure out what one group equals I will have the
square root
2 x 2 x 3 = 12
So √144 = 12
What if I can’t separate it into equal groups?
 In this case you do not have a perfect square.
 All you can do at this point is do your best to estimate what the
square root might be.
 To do this, you will need to identify the next closest perfect
squares above and below your number. Based on how close your
number is to either you can take your best guess at the decimal
value.
Example:
 √42 =
o If I quickly perform the prime factorization, I should find that
42 = 2 x 3 x 7
o I definitely can’t split this into two equal groups so I’m not
working with a perfect square meaning I’ll need to estimate
the square root.
o I can quickly square some numbers to find the perfect
square above and below 42.
62 = 36
72 = 49
o 42 is 6 above 36 and 7 below 49 meaning it’s not quite
halfway in between. As a result I will estimate the square
root is approximately 6.4.
 A quick check with my calculator tells me √42 = 6.48
so I would say 6.4 is a reasonable estimate.
How does this all relate to rational numbers?
 You are expected to be able to find or estimate the square root of
rational numbers.
 All this means is you would perform these steps on both the
numerator and denominator.
Examples:
 √
64
81
=
o Really this is asking me
√64
√81
o If I do the prime factorization on both numbers I’ll find…
64 = 2 x 2 x 2 x 2 x 2 x 2
81 = 3 x 3 x 3 x 3
o It appears I can split both of these up into two equal
groups…
64 = 2 x 2 x 2 x 2 x 2 x 2 = (2 x 2 x 2)(2 x 2 x 2)
81 = 3 x 3 x 3 x 3 = (3 x 3)(3 x 3)
o If I figure these both out I should find that √64 = 8 and
√81 = 9
o So √
 √
45
90
64
81
=
8
9
=
o If I do the prime factorization on both numbers I’ll find…
45 = 3 x 3 x 5
90 = 2 x 3 x 3 x 5
o It appears neither number can be split up into two equal
groups meaning I’ll need to estimate for both.
45 is about two thirds of the way between 36 (9 away) and
49 (4 away) so I’ll estimate its square root to be 6.7
90 is about halfway between 81 (9 away) and 100 (10 away)
so I’ll estimate its square root to be 9.5
o Based on all this I would estimate √
45
90
=
6.7
9.5
=
67
95
= 0.705
(we normally want to avoid decimals in fractions)
 A quick check on a calculator finds the real answer is
0.707
Let’s try a few together…
 √
16
 √
30
 √
22
81
85
𝟒
=
( )
=
( )
144
=
𝟗
𝟓𝟓
𝟗𝟐
(
𝟒𝟕
𝟏𝟐𝟎
)
Consider the following…
 What is the answer for
You are expected to be able to put together all this knowledge of
rational numbers, exponents and the order of operations.
Practice
Complete the Exponents and Rationals Practice handout.
Name:_______________ Date:_______________
Exponents and Rationals Practice
Use prime factorization to determine if the following numbers are
perfect squares. If they are, calculate their square roots. If they are not,
estimate their square roots.
1. 64
2. 196
3. 108
4. 24
5. 441
6.
7.
196
225
8.
78
110
9.
18
25
169
324