Unit 1: Topic 3: Simplification of Radicals

Competencies

·         Classify simplified radicals as rational or irrational

·         Distinguish between like terms and unlike terms

·         Use properties and theorems in simplifying radicals

Suggested ways of teaching this topic: Explanation by the Teacher and Practice of Students

Starter Activities

The teacher may start with some brainstorming questions like:

“When do we say that a radical is simplified or is in its simplest form?”

Expected Answers:

•         When the radicand has no square factors.

•         A radical is also in simplest form when the radicand is not a fraction.

Then, the teacher may explain “To put a radical expression in its simplest form, we make use of the following theorem:”

Theorem: For all non-negative real numbers a and b,

 

Here is a simple illustration: = 2  = 10

= = ·  = 3.

Therefore, we have simplified.

Let students practice the following problems themselves first!

a)    = 

b)    =  = = 5

c)    =  = = 3

d)    =  = 7

e)     =  = 4

f)    =  = 10

g)    =  = 5

h)    =  = 4

Example:  Reduce the following to their lowest terms.

a)

2

=

2

=

2

=

 

b)

3

=

3

=

3

=

2

 

c)

2

=

The radical is in its simplest form. The fraction cannot be reduced.

What are similar radicals?

Similar radicals have the same radicand.  We add them as like terms.

7 + 2 + 5 + 6 −  =  7 + 8 + 4.

2 and 6 are similar, as are 5 and −.  We combine them by adding their coefficients.

Lesson Notes

Examples: Simplify each radical, and then add the similar radicals.

a)         + = 3 + 2 = 5

b)         4 − 2 + = 4 − 2 +

= 4· 5 − 2· 7 +

= 20 − 14 +  =   7

 

c)   3 + − 2	=	3 + − 2
	=	3· 2 + 2 − 2· 4
	=	6 + 2 − 8
	=	2 − 2

 

d)   3 + +	=	3 + +
	=	3 + 2 + 3
	=	3 + 5

 

e)   1 − +	=	1 − +
	=	1 − 8 + 3
	=	1 − 5

Examples: Simplify the following.

a)	2	=	2		=	2 − ,			On dividing each term in the numerator by 2.
b)	5		=	5			=	2 +

 

c)

6

=

6

=

3

Dividing each term by 2.

Concluding Activities

Help students to conclude that we say that a radical is simplified or is in its simplest form when the radicand:

•         Have no square factors.

•         Is not a fraction

Make sure that students are capable of classifying simplified radicals as rational or irrationals, distinguishing between like terms and unlike terms and they can use properties and theorems in simplifying radical expressions allowing them to practice more. 

Practice Exercises

1.                           Simplify  

a)	b)	c)	d)
e)	f)	g)	h)

2.                           Simplify

a)                          

b)                          

c)                          

d)                         

e)                          

f)                           

g)                          

h)