Radical
Expressions &
Equations
Math 20 – Pre-Calculus
Chapter 5
Name:__________________
Class:___________________
1
Algebra and Number
Specific Outcomes
Solve problems that involve
operations on radicals and radical
expressions with numerical and
variable radicands.
General Outcome:
Develop algebraic reasoning and number sense.
Achievement Indicators:
The following set of indicators may be used to determine whether
students have met the corresponding specific outcome
2.1
2.2
2.3
2.4
2.5
2.6
2.7
Compare and order radical expressions with numerical radicands in
a given set.
Express an entire radical with a numerical radicand as a mixed
radical.
Express a mixed radical with a numerical radicand as an entire
radical.
Perform one or more operations to simplify radical expressions with
numerical or variable radicands.
Rationalize the denominator of a rational expression with monomial
or binomial denominators.
Describe the relationship between rationalizing a binomial
denominator of a rational expression and the product of the factors
of a difference of squares expression.
Explain, using examples, that
x
2.8
2.9
Solve problems that involve radical
equations (limited to square roots)
3.1
3.2
3.3
3.4
3.5
2
 x2 ,
2
x  x , and x 2   x; e.g., 9  3.
Identify the values of the variable for which a given radical
expression is defined.
Solve a problem that involves radical expressions
(It is intended that the equations will have no more than two radicals.)
Determine any restrictions on values for the variable in a radical
equation.
Determine the roots of a radical equation algebraically, and explain
the process used to solve the equation.
Verify, by substitution, that the values determined in solving a
radical equation algebraically are roots of the equation.
Explain why some roots determined in solving a radical equation
algebraically are extraneous.
Solve problems by modelling a situation using a radical equation.
2
Big Ideas:
Students will understand …

Radical numbers allow us to use exact values in real life situations like
measurement, distance and surveying. Radical number operations are the
fundamental skills useful in other mathematical topics like trigonometry
and coordinate geometry.
By the end of the unit students should:




That operations performed on radicals are similar to other number
systems and algebraic operations.
Radicals with even indices are limited to non-negative radicands, while
odd indices have no restrictions on the radicands.
Solving radical equations can yield extraneous roots.
There are conventions for simplifying answers after performing radical
operations.
3
5.1 Working with Radicals
Part I: Simplifying Radicals
Parts of a Radical
23 5
Like Radicals . . .
Restrictions on variables in radicals . . .
State the restriction on the following radicals . . .
x
x 5
3 x
x2
Recap from Math 10C:
How to Converting from a Radical to a Mixed Radical
and Vice Versa
Mixed Radical:
Entire Radical:
4
Convert Entire Radicals to Mixed Radicals
(a)
45
(b)
75
(c) 18
(d)
80
(e)
98
(f)
(h)
(g)
3
250
(i)
4
32a8b
3
32
20a2b4
Convert Mixed Radicals to Entire Radicals
(a) 2 18
(b) 2 3 5
(c) 5x 7 x2
(d)
5
1
10cd 3
2
Order Irrational Numbers
Arrange in order from least to greatest without the use of a
calculator
2 6,
3 3
(a) 5 ,
4 2,
(b)
3 5 ,
(c)
33 4 ,
 31 ,
43 2 ,
2 7 ,
23 5 ,
3
4 2
87
6
Part II: Adding and Subtracting Radicals
Recall
Simplify 3x – 2y + x + 4y by collecting “like terms”
Problem #1
Simplify 3 5  2 3  5  4 3 by collecting “like radicals”.
Example #1
Simplify the following:
(a) 7 6  3 6  6
(b) 2 3  2  3 3  4 2
(c) 2 x  5 y  4 y  x
(d)
1
2
7
7
4
3
Problem #2
Simplify by writing as simplified radicals first, then adding the coefficients of like
radicals
(a)
27  32  12  50
7
Example #2
Simplify the following:
(a)
24  54
(b) 6 45  5 80
(c)
27  45  2 75  3 80
(d) 2 98  10  5 8  3 40
8
(e)
(f)
3
250  3 3 128
2
2
1
1
125 
27 
45 
48
5
3
3
2
(g) 5 3x  27 x  2 12 x
Textbook
Page 278 #1-6, 8-12, 14,15,17
9
5.2 Part 1: Multiplying Radicals
Recall: Simplify 2 x 2  3xy
Steps for Multiplying Radicals
Multiplying Monomial Radicals
Example #1
(a)
3 5
(b) 2 3  5 6
(c) ( x 2)( x 6)
Multiplying Polynomial Radicals
Example #2
(a)
2(5  2 3)
(b)
2 6(2 3  10)

(d) 3 3 2 x3 2 3 3x  3 8 x 2
(c) 3 8c ( 12c  16)
10

Multiplying a Binomial  Binomial
Example #3
(a) ( 3  5)( 7  5)

(b) 3 2  2 5
(c)
10r  4
3

2
4r
2
3
6r 2  3 3 12r

Assignment: Page 289 #1-5
11
5.2 Part 2: Dividing Radicals
Recall: Simplify
12 x 2
4 x3
Steps for Dividing Radicals
Dividing Monomial Radicals
Example #1
(a)
(c)
2 15
(b)
8 3
42t 3
6t
(d)
12
25 15m4
5 3m4
169
64
In general, fractions should not be left with radicals in the denominator. The
technique used to write fractions in another form is called rationalizing the
denominator.
How to rationalize a MONOMIAL radical denominator . . .
Example #2
Rationalize the denominator in the following expressions.
(a)
(b)
4
5
3 6  15
3
3
(c)
(d)
9 27b2
23 16b
3 24  4 5
8
13
Rationalizing a Binomial Denominator
How to rationalize a BINOMIAL radical denominator . . .
Example #1
State the conjugate of the following:
(a) 2  3
(b)
Problem #1:
Rationalize the denominator of:
1
(a)
2 3
(b)
Example #2
Rationalize the denominator of:
(a)
(b)
52 3
3
5b  1
3 2
2  5t
Homework
Textbook Page 290 # 6-11,14,15,17
14
14 6
3 2 3
5.3 Radical Equations
Restrictions On Values for the Variable in a Radical Equation
Example #1
State the restrictions on the values for the variable in each radical equation.
a) y 
x5
c) y  2 x  5
b) y   2 x  3
d) y  3x
e) y  4  x  6
Solving Radical Equations Algebraically
Example #2
Solve the following radical equations. Check for extraneous solutions.
a)
3
2x  3
b) 6  3x
15
c) 0  3x  2  2
e)
2x2 
15 x  25
d) 3  x  7
f)
16
4 x2  5  x  2
g) 20   4 2 x  1  8
h)
4x  5  2x 1  2 .
Homework
Textbook Page 300 # 1-10,16
17
18