Download Section 7-2 Multiplying and Dividing Radical Expressions

Algebra 2
Multiplying, Dividing, Rationalizing and Simplifying…
Section 7-2
Multiplying and Dividing Radical Expressions
Lesson 7-2
Objectives
Algebra 2
• To multiply radical expressions
• To simplify radical expressions
• To multiply and simplify radical
expressions
• To find roots of quotients
• To divide radical expressions
• Solve algebraic equations
involving radical expressions
Multiplying and Dividing Radical Expressions
Lesson 7-2
Objectives
Algebra 2
• To multiply radical expressions
• To simplify radical expressions
• To multiply and simplify radical
expressions
• To find roots of quotients
• To divide radical expressions
• Solve algebraic equations
involving radical expressions
Multiplying and Dividing Radical Expressions
Lesson 7-2
Algebra 2
Property
For any nonnegative numbers a and b,
and any natural-number index k ,
k a k b  k ab.
Multiplying and Dividing Radical Expressions
Lesson 7-2
Algebra 2
Multiply
x2
3
4
19
3
x2  x 4
2
5  20
3
7  133
Multiplying and Dividing Radical Expressions
Lesson 7-2
Algebra 2
Factor and Simplify
Radical Expressions
k
ab  a
k
k
b
Multiplying and Dividing Radical Expressions
Lesson 7-2
Algebra 2
Simplify
20  4  5  2 5
50  25  2  5 2
3
32  8  4  2 4
3
3
3
Multiplying and Dividing Radical Expressions
Lesson 7-2
Algebra 2
Simplify
2x  4x  2
2
 2  x  2 x  1  2  ( x  1)
2
 2  x 1
2
Multiplying and Dividing Radical Expressions
Lesson 7-2
Algebra 2
Additional Examples
Multiply. Simplify if possible.
a.
b.
c.
3•
12
3•
12 =
3
–16 •
3
4
3
–16 •
3
4=
–4 •
16
3 • 12 =
3
36 = 6
–16 • 4 =
3
64 = –4
The property for multiplying radicals does not apply.
–4 is not a real number.
Multiplying and Dividing Radical Expressions
Lesson 7-2
Algebra 2
Additional Examples
Simplify each expression. Assume all variables are positive.
50x5
a.
50x5 =
=
52 • 2 • (x2)2 • x
Factor into perfect squares.
52 • (x2)2 •
n
= 5x2
b.
2•x
2x
54n8
3
54n8 =
3
33 • 2 • (n2)3 • n2
=
3
33(n2)3 •
3
2n2
n
b =
n
ab
definition of nth root
3
= 3n2
a•
3
2n2
Factor into perfect cubes.
n
a•
n
b =
n
ab
definition of nth root
Multiplying and Dividing Radical Expressions
Lesson 7-2
Algebra 2
Additional Examples
Multiply and simplify
are positive.
3
25xy8 •
3
3
25xy8 • 5x4y3
=
3
53x3(y3)3 • x2y2
=
3
53x3(y3)3 •
= 5xy3
3
x2y2
25xy8 •
n
5x4y3 =
3
3
x2y2
a•
3
n
5x4y3 . Assume all variables
b =
n
ab
Factor into perfect cubes.
n
a•
n
b =
n
ab
definition of nth root
Multiplying and Dividing Radical Expressions
Lesson 7-2
Algebra 2
Property
For any natural number index k and any
real numbers a and b, (b  0),
k
k
where a and b are real numbers,
k
a
a
k
k
b
b.
To find the k th root of a quotient, find the k th root of the numerator and
denominator separately.
Multiplying and Dividing Radical Expressions
Lesson 7-2
Algebra 2
Dividing Radical Expressions
To divide radical expressions
with the same index, divide the
radicands. Simplify if possible.
k
k
a k a

b
b
Multiplying and Dividing Radical Expressions
Lesson 7-2
Dividing Variables w/ Exponents
Algebra 2
• Reminder: When Multiplying
Expressions with similar Bases the
Exponents ADD!
• When Dividing Expressions with
similar Bases the Exponents
SUBTRACT!
3
3
5
32
2
3 4
3


2

16
3
3 1
2
2
Multiplying and Dividing Radical Expressions
Lesson 7-2
Algebra 2
Additional Examples
Divide and simplify. Assume all variables are positive.
a.
3
=
=
–81
3
3
3
b.
3
=
–81
3
=
192x8
3
3
3x
192x8
3x
=
3
–27
=
3
64x7
=
3
(–3)3
=
3
43(x2)3 • x
= –3
= 4x2
3
x
Multiplying and Dividing Radical Expressions
Lesson 7-2
Algebra 2
27
3

125
3
16x

4
y
3
5
3
27

3
125
16 x
y
2
2
y
x
2

4x
y
x
2
Multiplying and Dividing Radical Expressions
Lesson 7-2
Algebra 2
5
2 6
64a b
5
7
2a b

2 6
5
64a b
7
2a b
1
5 5
5
5
 32 5 b
a
2
64
a
5
5
7
2 a
6
5
b
b
1
2b
2 b
a
a
Multiplying and Dividing Radical Expressions
Lesson 7-2
Algebra 2
Rationalizing the Denominator
Writing the result without radicals in
the denominator.
Multiply by a 1 to make the
denominator a perfect power.
Multiplying and Dividing Radical Expressions
Lesson 7-2
Algebra 2
Additional Examples
Rationalize the denominator of each expression. Assume that
all variables are positive.
Method 1:
a.
3
3
5
5
=
3
5
Rewrite as a square root of
a fraction.
=
3•5
5•5
Then make the denominator
a perfect square.
=
15
52
=
15
5
Multiplying and Dividing Radical Expressions
Lesson 7-2
Algebra 2
Additional Examples
(continued)
Method 2:
a.
3
3
5
5
=
3•
5•
=
15
5
5
5
Multiply the numerator and
denominator by 5 so the denominator
becomes a whole number.
Multiplying and Dividing Radical Expressions
Lesson 7-2
Algebra 2
1
2
3
6
Multiplying and Dividing Radical Expressions
Lesson 7-2
Algebra 2
3
3
x
1
3
3
Multiplying and Dividing Radical Expressions
Lesson 7-2
Algebra 2
Additional Examples
(continued)
b.
c.
x5
3
5
4y
3
5
4y
3x2y
x5
3x2y
=
=
=
=
x5 •
3x2y
3x2y •
3x2y
3x7y
3x2y
x3
x
3xy
2
3x y
3xy
3y
=
3
5 • 42y2
4y • 42y2
=
3
80y2
=
=
43y3
2
3 10y2
4y
3 10y2
2y
Rewrite the fraction
so the denominator
is a perfect cube.
Multiplying and Dividing Radical Expressions
Lesson 7-2
Additional Examples
Algebra 2
The distance d in meters that an object will fall in t seconds is
given by d = 4.9t 2. Express t in terms of d and rationalize the
denominator.
d = 4.9t 2
d
t 2 = 4.9
t=
=
=
d
4.9
d • 10
49
10d
7
Multiplying and Dividing Radical Expressions
Lesson 7-2
Algebra 2
Multiply and Simplify
15
6
90  9  10  3 10
3 25 2 5  6 125  6  5  30
3
x 1
3
x 1 
3
( x  1)  x  1
2
Multiplying and Dividing Radical Expressions
Lesson 7-2
Algebra 2
Multiply and Simplify
3
x
2
3
4
x
( x  5)
3
8x
2
3x y
3 4
2x
2
2
8x y
4
Multiplying and Dividing Radical Expressions
Lesson 7-2
Algebra 2
Multiply and Simplify
3
18 y
3
3
4x
2
 72 x y  72  x  y
2
3
3
3
3
2
3
3
 8  9  x  y  2 y 9x
3
3
3
2
3
2
Multiplying and Dividing Radical Expressions
Lesson 7-2
Algebra 2
Do I have to Multiply and Simplify…Can’t I just Factor then Simplify?
18 y 14 y  252y
9 2 7 2 y
2
2
18 y 14 y
9 2 7 2 y
9 4 7 y
3 2 7 x
2
2
x
2
3
4
5
6
7
8
9
10
x2
4
9
16
25
36
49
64
81
100
x3
8
27
64
125
216
343
512
729
1000
x4
16
81
256
625
x5 Algebra 2
32
243
Algebra 2
Homework
Page
Simplify
Algebra 2
Multiply and Simplify
Algebra 2
3
3
0.3x 0.09 x  0.027x
20  5  4  2 5
3 3
54x
3
2
8
 27  2  x  x  x
3 2
23
2
3
 3 2  x  x  x  3x 2 x
3
3
3
3
3
3
3
2
Algebra 2
22.
4
80  16 5  2 5
4
4
4
Algebra 2
25.
3
(x+y)
4
 (x+y)  3 (x+y)  ( x  y ) 3 (x+y)
3
3
Algebra 2
28.
5 10 
50  25  2  5 2
Algebra 2
31.
3
2x y 12 xy
 24x y  4  6  x  y 
4
2
4
 2 6 x y  2x y 6
2
2
2