Unit 1: Topic 4: Rationalization of Radicals

Competencies

·         Explain the notion of rationalization.

·         Identify a rationalizing factor for a given expression.

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

Starter Activities

The teacher may start with revising the rules for multiplying and dividing radicals using examples:

The rules are:

1.      

2.

Let students do the problems themselves first then the teacher might summarize.

Example 1:   Multiply.

a)	·  =		b)  2· 3 = 6
c)	·  =	= 6	d)  (2)² = 4· 5 = 20

 

e)

=

The difference of two squares

Example 2:   Multiply, and then simplify:

a)       

 

b)      ( + )( − ).

Solution:   

1.     

 

2.      The student should recognize the form those factors will produce.

The difference of two squares

Practice and Consolidation is needed here! Thus, the teacher should guide students to recognize the forms of “difference of two squares” letting them practice on products of the following type.

Lesson Notes

Example 3:  Multiply.

a)   (+ )( − )  =  5 − 3 = 2

b)   (2+ )(2 − )  =  4· 3 − 6 = 12 − 6 = 6

c)   (1 + )(1 − )  =  1 − (x + 1)  =  1 − x − 1  =  −x

d)   (+ )( − )  =  a − b

Example 4:   Given (x − 1 − ) (x − 1 + )

a)   What form does that produce?

Expected Answer:

The difference of two squares of the form a²- b², where (x– 1) is "a" and is "b"

b)  Find the final product.

Example 5:   Multiply.

 

 

Example 6:   Simplify the following.

a)

=

b)

8

=

3 4

c)

=

a

=

a· a

=

a²

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What are Conjugate pairs?

The conjugate of a +  is a −.  They are a conjugate pair.

Example 7:   Multiply 6 −  with its conjugate.

Solution:  The product of a conjugate pair is the difference of two squares

(6 − )(6 + )  = 36 − 2 = 34

CONCLUSION

When we multiply a conjugate pair, the radical vanishes and we obtain a rational number.This process is called Rationalization.

Example 8:   Multiply each number with its conjugate.

1. x +     = = x² − y
2. 2 −    = (2 − )(2 + ) = 4 − 3 = 1
3.  +

This time, students should be able to write the product immediately:  6 − 2 = 4

4. 4 −   = 16 − 5 = 11

Example 9:   Rationalize the denominator of

Solution:  

Multiply both the denominator and the numerator by the conjugate of the denominator; that is, multiply them by 3−.

1

=

9 − 2

=

7

The numerator becomes 3−.  The denominator becomes the difference of the two squares.

Example 10: Rationalize the denominator of

Solution:		=	3 − 4	=	−1
				=	−(3 − 2)
				=	2 − 3

Example 11:   Write out the steps that show each of the following equivalences.

a)

1

=  ½()

 

1

=

5 − 3

=

2

=

½( −)

 

b)

2     3 +

=  ½(3 − )

 

2     3 +

=

9 − 5

=

4

=

½(3 − )

 

c)

_7_     3 +

=

6

 

	_7_     3 +	=	9· 5 − 3	=	42	=	6

 

d)

− 1

=

3 + 2

 

− 1

=

2 − 1

=

2 + 2 + 1,

 

Example 12:    Simplify

At every step of the solution the teacher is expected to ask the students to give reasons.

Solution:		=		Why?
		=		Why?
		=	6 − 5	Why?
		=	6 − 5	Why?	.

 

Example 13:    Simplify

 

		=			Why?
		=			Why?
		=	3 − 2		Why?
		=	3 + 2		Why?

Concluding Activities

Make sure that the students summarize that rationalization of the denominator can be done by multiplying both the denominator and the numerator by the conjugate of the denominator. Check if every student has developed the skill of rationalizing radical expressions by giving them practice exercises.

Practice Exercises

1.      Rationalize the following by multiplying with an appropriate conjugate and simplify

a)        

b)        

2.      Rationalize the denominator and simplify

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