Radicals
I am not a teacher; only a fellow
traveler of whom you asked the way.
George Bernard Shaw (1856-1950)
British dramatist, critic, writer.
Squaring a Number
42 means 4 x 4 = 16. It is called 4 squared and
16 is called a square number.
A square with sides that
are 4 long has an area of
16 units2
42 = 16
You can see the reason for
calling it 4 squared and
why 16 is a square
number
Perfect Squares
The numbers 16, 36, and 49 are examples of
perfect squares. A perfect square is a number
that has integers as its base. Other perfect
squares include 1, 4, 9, 25, 64, and 81.
Number
Square
Number
Square
1
2
3
4
5
1
4
9
16
25
6
7
8
9
10
36
49
64
81
100
Square Roots
Many mathematical operations have an inverse,
or opposite, operation. Subtraction is the
opposite of addition, division is the inverse of
multiplication, and so on. Squaring has an
inverse too, called "finding the square root."
Square Roots
Remember, the square of a number is that
number times itself.
The square root of a number, n, written
n
is the number that gives n when multiplied by
itself. For example,
100  10
because 10 x 10 = 100
Square Roots
√16 means ‘the square root of
16’ and √16 = 4
A square with an area of 16
has sides that are 4 units long.
Taking the square root of a number is the
reverse process of squaring the number.
Square
root
1 = 1
Perfect
square
81
Square
root
81 = 9
4
4 = 2
100
100 = 10
9
9 = 3
121
121 = 11
16
16 = 4
144
144 = 12
25
25 = 5
169
169 = 13
36
36 = 6
196
196 = 14
49
49 = 7
225
225 = 15
64
64 = 8
Perfect
square
1
Square Roots
Every positive number has two
square roots, one positive and one
negative. One square root of 16 is 4,
since 4 × 4 = 16.
The other square root of 16 is –4,
since (–4) × (–4) is also 16.
You can write the square roots of 16
as ±4, meaning “plus or minus” 4.
Square Roots
The radical symbol
returns only the positive root.
This is called the principal square root of the
number.
To get the negative root, simply take the opposite
of the principal root:
25  5
 25  5
What about negatives?
Consider a negative
under the radical
 25
This is asking ‘what
number multiplied by
itself returns a -25?’
5×5 = 25 positive
(-5) ×(-5) = 25 positive
No product returns a
negative value
Example
A square shaped kitchen table has an area of 16
square feet. Will it fit through a van door that has a
5 foot wide opening?
Find the square root of 16 to find the width of
the table. Use the positive square root; a
negative length has no meaning.
16  4
So the table is 4 feet wide, which is less than 5 feet, so
it will fit through the van door.
Estimating Square Roots
Finding square roots of numbers that aren't
perfect squares without a calculator
1. Estimate - first, get as close as you can by finding
two perfect square roots your number is between.
2. Divide - divide your number by one of those square
roots.
3. Average - take the average of the result of step 2
and the root.
4. Use the result of step 3 to repeat steps 2 and 3
until you have a number that is accurate enough for
you.
Estimating Square Roots
Calculate the square root of 10 without a
calculator
10
1. Estimate - first, get as close as you can by finding
two perfect square roots your number is between.
9
10
16
3
?
4
10
Lies between 3 and 4
Estimating Square Roots
Calculate the square root of 10
10
2. Divide - divide your number by one of those square
roots.
3  10  4
10
= 3.33
3
Divide 10 by 3.
(you can round off your answer)
Estimating Square Roots
Calculate the square root of 10
10
3. Average - take the average of the result of step 2
and the root chosen in step 2.
3.33 + 3
= 3.1667
2
Average 3.33 and 3.
Estimating Square Roots
Calculate the square root of 10
10
4. Use the result of step 3 to repeat steps 2 and 3
until you have a number that is accurate enough for
you.
Repeat step 2 with 10 and 3.1667
10
= 3.1579
3.1667
3.1667 + 3.1579 = 3.1623
Repeat step 3 with
3.1667 and 3.1579
2
Estimating Square Roots
Calculate the square root of 10
10
Try the answer →
Is 3.1623 squared equal to 10?
3.1623 x 3.1623 = 10.0001
If this is accurate enough for you, you can stop!
Otherwise, you can repeat steps 2 and 3.
10  3.1623
Multiplying Radicals
Consider the following product
4 9 = 2 × 3
=6
Another way:
4 9
 4 9
 36
=6
Rules of Radicals
This process leads to a few simple rules we
can use with radicals
 m
2
1.
m 
2.
m  n  m n
3.
m

n
2
m
n
m
Multiply
5  2  5 2
 10
3  3  3 3
 9
3
Simplifying Radicals
There are three components to a simplified
radical:
1. All perfect square factors should be
removed from the radical
2. All fractions should be removed from the
radical
3. All radicals should be removed from the
denominator
Simplify Radicals
Recall we prefer to simplify fractions by
removing common factors to make the
denominator smaller.
For radicals, we will follow a similar process,
except we will concentrate on perfect square
factors
Perfect
Square
8  4 2
 4 2  2 2
Simplify
Think of the
largest perfect
square that
divides into 32
4 yes → 4×8
9 no
16 yes → 16×2
32 
16 
4 2
2
Simplify
45 
Think of the
largest perfect
square that
divides into 32
4 no
9 yes → 9×5
9

3 5
5
Simplify
Even if you can’t
think of the
largest perfect
square, you can
always simplify
down through
each perfect
square:
4 6
96 
4 
24
4 
6
2 2 6
Simplify
Use the division
property
Check:
2/3 ͯ 2/3 = 4/9
4
4

9
9
2

3
Simplify
Use the division
property to write
as single
fraction
Simplify the
fraction
Use the division
property again
8
8

50
50
4

25
4 2


25 5
Simplify
Recall:
All radicals should be
removed from the
denominator
1
3
How do we remove the radical
from the denominator?
Since this is a fraction, then let’s think
about how we change the denominator
of a fraction?
(Without changing the value of
the fraction, of course.)
1
3
We multiply the denominator and the
numerator by the same number
1 1 3 3


4 4  3 12
How can we change the radical
value to a rational value?
Simply multiply the radical by itself!
2
3  3
3  3
Remember when we square a
square root, the radical goes away
 m
2
m
In our fraction, to get the radical
out of the denominator, we can
multiply numerator and
denominator by
3
1
1 3
3


3
3 3
3
Because we are changing the
denominator to a rational number,
we call this process
rationalizing.
3
1

3 3
Simplify
Use the division
property
Rationalize the
denominator
3
3

5
5
3
5
15



5
5
5
Exponents under the Radical
Variables with exponents may exist under the
radical, but the same simplifying process still
applies
All even exponents
are perfect squares
Notice the pattern
half the exponent
x  x 
2
2
 
 x 
 x 
x  x
4
x
6
x
8
2 2
3 2
4 2
4x
2
Exponents under the Radical
To simplify, we can use the perfect square
property
2
m m
Rewrite as a perfect
square
10
half of
10 is 5
x

x 
5 2
No more
radical
x
5
The radical and the square
offset each other
Simplify
4x
2
12
x
( x  1)
8
 2x
Half of the
exponent
x
6 is half of 12
6
 ( x  1)
4
4 is half
of 8
Exponents under the Radical
Odd exponents can be rewritten as an even
and an odd by using the rules of exponents
Rewrite odd
exponent as one
3
2
3=2+1
x
x x
less (even) plus
one
8
9
x x
x
9=8+1
x
15
x x
15 = 14 + 1
x
101
x
101 = 100 + 1
14
100
x
Simplify
Rewrite odd exponent as
one less (even) and one
half of 6 is 3
x
7
 x x x
6
3
the other x stays
under the radical
x
Adding Radicals
Adding radicals is same as adding like terms:
1+1=2
x + x = 2x
+=2
3 3 2 3
Adding and Subtracting Like Radicals
Add or subtract, as indicated. Assume all
variables represent positive real numbers.
7 2  8 18  4 72
Cannot add yet.
Simplify to see if they
are like radicals
7 2  8 18  4 72  7 2  8 9  2  4 36  2
 7 2  83 2  46 2
 7 2  24 2  24 2
7 2
Combined Operations with
Radicals


You follow the same
steps with these as
you do with
polynomials.
Use the distribution
property.

Example:
2 ( 10  2 )
20  4
2 52