Calculus and Vectors – How to get an A+
1.1 Radical Expressions: Rationalizing Denominators - Handout
Ex 1. Simplify:
A Radicals
a a =a
n
a)
( a) = a
n
m
( a )=
m
an
3 3
b) ( 5 ) 3
n
= (n a ) m
Note: If n is even, then a ≥ 0 for
n
c) (3 7 ) 5
a.
B Rationalizing Denominators (I)
a
a
c a c
=
=
bc
b c b c c
Ex 2. Rationalize:
2
C Conjugate Radicals
a+ b ⇔a− b
Ex 3. For each expression, find the conjugate radical.
3 5
a) 2 + 3
⇒
a+ b⇔ a− b
b)
2− 3
a +b c ⇔ a −b c
c)
3+2 5
a b +c d ⇔a b −c d
D Difference of squares identity
(a + b)(a − b) = a 2 − b 2
d) 2 5 + 3 7
⇒
⇒
⇒
Ex 4. Use the difference of squares identity to simplify:
a) (a + b )(a − b )
b) ( a + b )( a − b )
c) ( a + b c )( a − b c )
E Rationalizing Denominators (II)
Hint: Multiply and divide by the conjugate radical of the
denominator.
Ex 5. Rationalize the denominator:
3
a)
1− 2
4
b)
2+3 5
2
c)
3− 6
F Rationalizing Numerators
Hint: Multiply and divide by the conjugate radical of the
numerator.
Ex 6. Rationalize the numerator:
5− 3
G Equivalent Expressions
Hint: You may get equivalent expressions by
rationalizing the numerator or denominator.
Note: State restrictions.
Ex 7. Find equivalent expressions by rationalizing.
State restrictions.
x −1
a)
x −1
2 −1
b)
x+9 −3
x
1.1 Radical Expressions: Rationalizing Denominators - Handout
© 2010 Iulia & Teodoru Gugoiu - Page 1 of 2
Calculus and Vectors – How to get an A+
1
c)
H More algebraic identities
3
3
2
2
a − b = (a − b)(a + ab + b )
a 3 + b 3 = (a + b)(a 2 − ab + b 2 )
a 4 − b 4 = (a − b)(a + b)(a 2 + b 2 )
x+h
h
−
1
x
Ex 8. For each case, the numerator and denominator
have a common zero. Use algebraic identities to
eliminate the common zero. State restrictions.
x −1
a)
3
x −1
b)
x4 −1
x3 − 1
Reading: Nelson Textbook, Pages 6-8
Homework: Nelson Textbook: Page 9, #1a, 2a, 3a, 4a, 5, 6a, 7ac
1.1 Radical Expressions: Rationalizing Denominators - Handout
© 2010 Iulia & Teodoru Gugoiu - Page 2 of 2