MCR3U1
Day 6: Radicals
Date _______________
Chapter 3: Quadratic Expressions
RADICALS
What is a radical?
An expression that has a square root, cube root, etc.
The symbol is
Types of Radicals
An entire radical is a radical with a coefficient of 1
(e.g.
)
A mixed radical has a coefficient other than 1
(e.g.
).
A. SIMPLIFYING (Reducing) RADICALS
To simplify means to find another expression with the same value. It does not mean to find a
decimal approximation.
METHOD 1: LARGEST PERFECT SQUARE
METHOD 2: PRIME FACTORS
1. Find the largest perfect square which will
1. Factor out the number into its prime factors.
divide evenly into the number under your radical
sign.
48
24
12
6
3
1
Dividend
Divisor
48 16
the largest perfect square
3 3
1
that divides evenly into 48 is 16
2. Write the number appearing under your
radical as the product (multiplication) of the
perfect square and your answer from dividing.
3. Give each number in the product its own
2
2
2
2
3
48=2x2x2x2x3
2. Write all the prime factors under your radical
3. Give each twin numbers and single numbers in the
radical sign.
product their own radical signs
4. Reduce the "perfect" radical which you have
4. Reduce the "perfect" radical which you have now
now created.
5. You now have your answer.
created
5. You now have your answer
1
MCR3U1
Day 6: Radicals
Date _______________
Chapter 3: Quadratic Expressions
i) Simplify the following “entire” radicals.
a)
b)
c)
d)
ii) Express each of the following as “entire” radicals.
a)
b)
B. MULTIPLYING/ DIVIDING RADICALS
When multiplying radicals, you must multiply the numbers OUTSIDE (O) the radicals AND then
multiply the numbers INSIDE (I) the radicals.
such as
When dividing radicals, you must divide the numbers OUTSIDE (O) the radicals AND then divide
the numbers INSIDE (I) the radicals.
such as
Rationalizing The Denominator
If a radical appears in the denominator of a fraction, it will need to be "removed" if you are trying
to simplify the expression. To "remove" a radical from the denominator, multiply the top and
bottom of the fraction by that same radical to create a rational number (a perfect square radical)
in the denominator. This process is called rationalizing the denominator.
Simplify
Answer
2
MCR3U1
Day 6: Radicals
Date _______________
Chapter 3: Quadratic Expressions
Multiply or divide, then simplify the following radicals
a)
b)
d)
c)
e)
D. ADDING RADICALS
When adding or subtracting radicals, you must use the same concept as that of adding or
subtracting "like" variables. In other words, the radicals must be the same
before you add (or subtract) them.
Like Radicals
example
Ex1: Add
Ex2: Add
non-example
Since the radicals are the same, simply add the numbers in
front of the radicals (do NOT add the numbers under the
radicals).
Answer:
Since the radicals are not the same, and both are in their simplest
form, there is no way to combine these values. The answer is the
same as the problem.
Answer:
Add the following radicals
a)
b)
3
MCR3U1
Day 6: Radicals
Date _______________
Chapter 3: Quadratic Expressions
MULTIPLYING BINOMIALS
N
1. Simplify. Express your answer as a radical in simplest form.
a) 24  18
b) 7  8
c) 11  14
e)
20  18
f)  32  72
g)
20  32  18
2. Simplify. Express your answer as a radical in simplest form.
a) (3  2 )(3  2 )
b) (7  7 )(7  7 ) c) ( 11  x )( 11  x )
e) (2  6 )(3  10 )
1a) 12 3
f) (1  15 )(8  15 )
b) 2 14
c) 154
g) (x  1  2)(x  1  2 )


d)
8   18
h)
24  54  18
d) ( 5  3)(3  5 )
h) (y  55)(y  22)
d)  12
e) 6 10
f)  48
g) 48 5
h) 108 2
2a) 7
b) 42
c) 11 – x2
d) –4
e) 6  3 6  2 10  2 15 f)  23  9 15 g) x 2  2x  1 h) y 2  ( 22  55)y  11 10
4
MCR3U1
Day 6: Radicals
Date _______________
Chapter 3: Quadratic Expressions
SIMPLIFYING RADICALS
1.
Simplify:
a) 12
d)
b)
450
2. Simplify:
12 2
a)
3
3. Solve for x.
a) x 2  169
e)
b)
20
72
6 6
6
b) 3x 2  144
c) 45
f) 200
c)
10 11
12
c) ( x  4) 2  81
100
144
d)
d) 2( x  1) 2  18
4. Express both roots in decimal form, rounded to 3 decimal places.
5
7 2 10
a) x  2  2 2
b) x  2 
c) x   
3
5
5
3 8  12
2
5. Simplify:
a) 2 2  27  2 12  8
b)
3
6  12
d)
2
6 10
c)
12
e) ( 2  3)( 2  3)
f) (3 2  4 3 )(3 2  4 3 )
15 6
h)
5 12
j) (6  4 3 )(6  4 3 )
g) 2 24 (3 3 )
i) 2 6 (3 2  3)
Answers
1. a) 2 3
d)
24 8
b) 2 5
c) 3 5
15 2
e) 6 2
6
5 11
6
2. a) 4 2
b)
3. a)  13
b)  4 3
c)  5 , 13
4. a) –0.828 , 4.828
b) –2.745 , –1.255
c) –2.665 , –0.135
5. a) 4 2  7 3
b) 8
c)
e) - 7
f) - 30
g)  36 2
i ) 12 3  6 2
j) - 12
c)
5
f) 10 2
d)
5
6
d) –4, 2
d) 3  3
3
h)
2
5
MCR3U1
Day 6: Radicals
Date _______________
Chapter 3: Quadratic Expressions
1.
2.
196 
3.
20 
4.
75 
5.
48 
150
6.
7.
8.
9.
10.
2 5 8 3 
4 2  3 14 
( 12 2 )  ( 5 6 ) 
3 57 5 
4 2  3 10 
11.
12.
13.
14.
15.
4 14

8 7

4 18

3 2

16.
18.
19.
14 12

7 24
17.
4 5

2 15
10 10

15 15
4
21.
22.
23.
24.
2 3  14 3 
3 83 2
8 3  4 27 
5 50  2 8  8 2 
12 6

3 2
5 60

2 3
3 150

5 3
20.
1
2
6
18
5
25.
48  4 27  2 75 
Answers (jumbled):
2 2
2 5
12 3
5 3
3
120 3
4 3
3 2
5 3
5 6
21 2
24 7
16 15
105
4 3
2
2
4
2 3
5 5
9 2
4 3
14
24 5
2
2 3
3
2 6
9
6