Transcript Operations with Radicals

Multiplying Binomial Radial
Expressions and Rationalizing
with Conjugates.
MA.912.A.6.2 Add, subtract, multiply, and
divide radical expressions (Square roots and
higher)
Multiplying Radical Expressions
(
5 4 - 20
)
= 5 · 4 - 5 · 20
= 4 5 - 100
= 4 5 -10
Distributive
Property
Product Property
Of Radicals
Simplify
Multiplying Radical Expressions
(
=
[
2- 5
] [
2· 2 +
2
)(
2 -3 5
)
] [
] [
]
2 · (-3 5) + (- 5) · 2 + (- 5) · (-3 5)
-3 10
- 10
17 - 4 10
+15
Multiplying Radical Expressions
(3 6 - 2 3)(4 + 3)
F
O
I
L
3 6 ×4 + 3 6× 3 - 2 3 × 4 - 2 3 × 3
12 6 + 3 18 -8 3 - 2 ×3
12 6 + 9 2 - 8 3 - 6
Since there are no like terms, you can not
combine.
Conjugate Binomials
The conjugate of
a + b is a - b
When you multiply conjugate binomials,
The product will be:
The Difference of Squares.
(a + b)( a - b) = a - b
2
2
Conjugate Binomials
The conjugate of
a b + c is a b - c.
(a
=a
2
)(
b +c a b -c
2
b -c
2
)
= a b-c
2
2
Multiplying Conjugates
(8 - 5 3)(8 + 5 3)
8× 8 + 8× 5 3 - 5 3 ×8 - 5 3 ×5 3
64 + 40 3 - 40 3 - 25× 3
64 - 75
-11
What other
method would
yield the same
answer with less
work?
Multiplying Conjugates
(8 - 5 3)(8 + 5 3)
2
=8 -5 3
= 64 - 25(3)
2
= -11
2
Rationalizing Using Conjugates
You should recall that a radical expression is
not considered simplified if there is a radical in
the denominator. The process of eliminating the
radical in the denominator is called rationalizing.
5
Rationalize :
2
5
2 5 2
·
=
2
2
2
Rationalizing Using Conjugates
When the denominator contains a binomial with a
radical, one must multiply by the conjugate in
order to rationalize the denominator.
Simplify :
c
a +b
Multiplying by
1 does not change
the value of the
number.
(
) (
)
c
a -b c a -b c a -b
·
=
=
2
2
2
a +b
a -b
a-b
a -b
NO RADICAL!
Rationalizing Using Conjugates
Simplify:
5+6
5-3
5+6 5+3
×
5-3 5+3
Multiply by the
conjugate.
Multiply numerators
Multiply denominators.
5 + 3 5 + 6 5 + 18
5-9
Combine like terms
Finished on next slide.
5 + 3 5 + 6 5 + 18 Combine like terms
5-9
Never leave
23 + 9 5
Negative in the
-4
Denominator!
Distribute