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10.3 Multiplication & Division of Radical Expressions
When multiplying radical expressions, you can just put
everything under the radical sign.
Multiply :
3 x 4 2 x 2 y 6 xy 2
= 3 x 4 ⋅ 2 x 2 y ⋅ 6 xy 2
= 36 x 7 y 3
= 36 x 6 ⋅ x ⋅ y 2 ⋅ y
= 6 x 3 y xy
Distributive Property:
(
2x x + 2x
)
= x 2 x + ( 2 x )( 2 x )
= x 2x + 2x
BE CAREFUL!
2x ≠ 2x
2 x should be written as x 2
to avoid confusion.
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Example 3 Multiply
(2
)(
x − y 5 x −2 y
)
Use FOIL
(2 x )(5 x )+ (2 x )(− 2 y )+ (− y )(5 x )+ (− y )(− 2 y )
= 10 x
− 4 xy
− 5 xy
+ 2y
= 10 x − 9 xy + 2 y
You try this one :
(
)(
Multiply 3 x − y 5 x − 2 y
)
Example 3 Multiply
(2
)(
x +7 2 x −7
( )( )
)
( )
( )
= 2 x 2 x + (− 7 ) 2 x + (7 ) 2 x + (7)(−7)
= 4 x − 49
Notice that this was the special factoring case of the difference of two squares :
(a + b)(a-b) = a 2 -b 2
( a + b) and (a-b) are called conjugates of each other.
Notice that when ra adical expression has two terms, all
radicals disappear when you multiply the expression by its
conjugate.
x + 1 x −1
Try this one:
(
)(
)
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Radical Expressions in Simplest Form
A radical expression is in simplest form if:
1. The radicand contains no factor greater than 1 that is a perfect
square.
2. There is no fraction under the radical sign.
3. There is no radical in the denominator of a fraction.
Quotient Property of Square Roots
4x2
z6
a
a
=
b
b
is not in simplest form because there is a fraction under the
radical sign. This can be simplified by taking the square root
of the numerator and the denominator.
4x2
4x2 2x
=
= 3
6
6
z
z
z
Simplify
4x2 y
4x2 y
=
xy
xy
= 4x
=2 x
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2
3
Is not in simplest form because there is a radical expression in
the denominator;
The way to simplify is to multiply both numerator and
denominator by
3
2
3 2 3
⋅
=
3
3 3
This doesn’t always work when there is a two-term expression with
at least one radical term added to another term.
2y
y +3
⋅
y +3
y +3
=
2y
(
)
y +3
( y) +6
2
y +9
The trick for these types is to multiply the numerator and
denominator by the conjugate.
2y
y +3
⋅
y −3
y −3
=
(
=
=
2y
(
y +3
)(
y −3
)
y −3
)
2y2 − 3 2y
( y) −3
2
2
y 2 − 3 2y
y −9
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SIMPLIFIED!
10.4 Solving Equations Containing Radical Expressions
Property of Squaring Both Sides of an Equation
If a and b are real numbers and a=b, then a2=b2
3x + 2 = 5
Solve :
3x = 3
Square both sides to get x out from under the radical sign.
( 3x )
2
= 32
3x = 9
x=3
Check :
3(3) + 2 = 5
9 +2=5
3 + 2 = 5 TRUE
It’s very important to check your solution because some
“solutions” actually make the original equation untrue.
Example 2B
Notice that when you get the constants on one side, your equation
says that the radical expression must equal a negative number.
This is impossible! Therefore there is NO SOLUTION to an
equation like this.
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Solve:
x − x −5 =1
In this case put the radical expression s on opposites of each other
before squaring both sides.
x =1+ x − 5
( x ) = (1 +
2
x −5
)
2
x = 1 + 2 x − 5 + ( x − 5)
This got rid of the radical on the left but we
still have a radical on the right. Combine like term
then start the process again.
x = 1 + 2 x − 5 + ( x − 5)
x − x = 1+ 2 x − 5 + x − x − 5
0 = −4 + 2 x − 5
(4 )2 = (2
x −5
16 = 4( x − 5)
)
2
16 = 4 x − 20
36 = 4 x
9= x
CHECK :
9 − 9−5 =1
3− 4 =1
3− 2 =1
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APPLICATIONS: Pythagorean Theorem
B (hypotenuse)
A
(leg)
B
(leg)
A +B =C
2
2
2
A2 + B 2 = C
Likewise;
A =C −B
2
2
2
A = C 2 − B2
B =C −A
2
2
2
B = C 2 − A2
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