# Expressions Containing Several Radical Terms

```Section 7.5 Expressions Containing




Products &amp; Quotients of Several Terms
Rationalizing Denominators (part 2)
Terms with Differing Indices
7.5
1



Each term must have a radical with identical
Law of distribution allows combining or
factoring
3 2 and 7 2

4
4
2 x 3 x and 11 3 x
3 5 and 2 3
5
4
2 y 3 x and 4 3 x
7.5
2
Simplify before Trying to Combine
2 12  3 48  3 3 
2 4 3  3 16 3  3 3 
2 ( 2 ) 3  3( 4 ) 3  3 3 
4 3  12 3  3 3   5 3
6 7 4 7 
(6  4) 7 
10
7
5
5
6 4x  3 4x 
5
( 6  3) 4 x 
5
9 4x 
3
3
3
9 54 3 
4x 
can ' t be simplified
4x 
4x
7.5
3
Use Distribution to
Multiply Like Indices


3 x
5 
x 3
3 5 
4
3
2


35 2 
4 9  20
6
4 ( 3 )  19
6  5(2) 
2  19

2
y 
3
2 
y
3
y 
3
3
y
3
2y
3
x 3  15

3
3
y
2
y
2 
6 5 4 
6
7.5
4
The Beauty of Square Root Conjugates



a 
( a) 
2
ab

Square roots only:

Middle term disappears
b

a 
ab 
7 3


7 3 
( 7) 3 7  3 7 9 
2

7  9  2
b 
ab  ( b ) 
2
2
3x 

5z 2 3x 

5z 
( 2 3 x )  2 15 xz  2 15 xz  ( 5 z ) 
2
2
4 ( 3 x )  5 z  12 x  5 z
7.5
5
Try this one in class:
2


3x  y 5 y 2 3x  y 5 y 
4 ( 3 x )  2 y 15 xy  2 y 15 xy  y ( 5 y ) 
2
12 x  5 y


2
3
Now this one:
3
a 
3
2
2
3
a a 
2
3

b
( a ) 
3
2
b
3
3
a 
2
a b 
2
3
3

b 
3
a b  ( b) 
2
2
2
7.5
6
Using Conjugates to Rationalize
Denominators (Part 2) – [or Numerators]
4

3x
4
2
5

2
4

( 3  x)
( 3  x) ( 3  x)
(4 
3 x
2
( 5
2)
2) ( 5 
2)
53  2
(4 
2)
1614
2
2 ) (4 
2)
2)
( 5

4 3  4x


4 54 2
10  2
Rationaliz e the numerator
4
5
2
2

(4 
( 5
2)

7.5

4 5
10  4 2  2
7
Terms with Differing Indices
7.5
8
More UnLike Indices
x 
3
3
x
3
2
 x x
 x
(  )
3
2
 x
11
6
 x
6
1
3
1
3
4
y 
5
4
5
3
y
 y y
 y
 y
x
5
 y
3
7
2
a b
7
3
ab
( 11 25  1228 )

2
3
2
3
1
2
1
2
a b
a b
43
12
a
3 12
2
y
( 23  12 )
b
( 23  12 )
7
1
6
a b

7.5
6
1
6
ab
9