1
Warm-up:
I. Exponents and Roots
1. Find the values of x n for different values of x and n.
x 1
x2
(1)2=
x2
1
x3
(1)3=
2
1
(1)4=
2
1
(2)4=
1
x5
(1)5=
(2)5=
1
9
(4)2=
(3)3=
16
(3)4=
27
(4)3=
2
(3)5=
(5)2=
64
81
(4)4=
243
256
25
(5)3=
125
(6)2=
(5)4=
(6)3=
625 1296
5
(4)5=1024
(5)5=3125
4
5
6
7776
2) (.3)2
3) (.4)2
4) (.5)2
.04
.09
.16
.25
5) (.2)3
6) (.3)3
7) (.4)3
8) (.5)3
.008
.027
.064
.125
9) (.2)4
10) (.3)4
11) (.4)4
12) (.5)4
.0016
.0081
.0256
.0625
Parts of a radical:
a
b
‘a’ is called the
‘b’ is called the:
INDEX
RADICAND
216
6
2. Then go back and find the nth root of each number.
Simplify:
1) (.2)2
36
6
5
4
3
x6
5
4
3
32
16
x5
4
3
2
1
(3)2=
x4
3
8
(2)3=
1
x4
4
(2)2=
1
x3
6
2
If no index is written, this implies an index of :
2
II. Finding Real Roots
18. 3
3
8x6  23 x6  2 x2
19.
5 243 y 5  5 35 y 5  3 y
22.
21. 4
256 x 4 y12
0.01
23.
(.1) 2
4 44 x 4 y12
4 xy3
.1
20.
1
1 1


81
92 9
3
0.001
3
(.1)3
.1
Find each real root.
15.
3 8 
27
24.
3 3
2
3 3
3
4
0.0081
4
(.3) 4
16.

2
3
27.
0.09
??? No Real Solution
0.3i {If complex ok}
16
24 2


81 4 34 3
25. 3
3
0.3
17. 4 625
4
4
27
No real roots
26.
(3)3
27
3
(3)4
3
3
28.
3
5
32
5
(2)5
29.
16
??? No real solution
4
2
Summarizing:
Even Roots
Odd Roots
Of a negative number
EVEN
 - #  NRS
ODD
 - #  () #
Of a positive number
EVEN
 +  #  ( ) #
ODD
 +  #  ( ) #
3
III. Practice with Roots and Variables.
28.
29. 3
125 x 6 y12
9 x6 y12
32 ( x3 )2 ( y 6 )2
3 x3 y 6
31.
5
d 20
32.
-d4
34. n 4n
x
x4
81a16b 20
3 ( 5)3 ( x 2 )3 ( y 4 )3
4 (3) 4 ( a 4 ) 4 (b5 ) 4
5 x 2 y 4
3a 4b5

 x  4 2
(x-4)
35.
30. 4
33. 3 x 2  6

6
(x2 + 6)2 = x4 + 12x2 + 36
3n 12n
x
x4
IV. What happens when the exponents aren’t multiples of the
index? (assume all variables are positive)
36.
27 x5 y12
37.
3 25 x 6 y8
33 x5 y12
3x2 y 6 3x
38.
4 27 x5 y11
3 32 x 6 y8
2 x2 y 2 3 2 y 2
39.
3 625 x5 y11
4 33 x 4 x1 y8 y3
3 (1)3 54 x5 y11
xy 2 4 27 xy3
5 xy3 3 5 x 2 y 2
4
Find the two real-number solutions of each equation.
40. x2 = 4
41. x4 = 81
+2
42. x2 = 0.16
x= + 3
x= + .4
43. x2 =
x=
16
49

4
7
44. A cube has volume V = s3, where s is the length of a side. Find the side length for a cube
with volume 8000 cm3.
S3 = 8000
S=
3
203
20
V. A twist? Simplify:
2 =2
45)
2
48)
51)
48)
3
4
5
46)
( 2) 2
=2
47)
( x)2
=|x|
(2)3
=2
49)
3
( 2)3
=-2
50)
3
( x )3
=x
(2) 4
=2
51)
4
( 2) 4
=2
52)
4
( x)4
=|x|
(2)5
=2
49)
5
( 2)5
=-2
50)
5
( x )5
=x
Summary:
even
variable =
| variable|
odd
variable = variable
Unless the instructions say: “do NOT assume all variables are positive,” you do not
need to worry about absolute values.
5
Find all the real cube roots of each number.
51. 343
-7
52. 0.064
53.
- 0.4
Find all the real fourth roots of each number.
54. 81
No real
55. 0.0001
0.1
1000
27
10
3
6
7
7.2 Multiply and Divide Rational Functions
1) Simplifying Radicals (assume all variables are positive):
3
a) √50𝑥 5
b) √54𝑥 8
3
52 2 x 4 x
c)
3321 x 6 x 2
3x 2 3 2 x 2
5x2 2 x
𝑛
𝑛
𝑛
𝑛
Rule: If √𝑎 and √𝑏 are real numbers, then √𝑎 × √𝑏 = √𝑎𝑏.
a)
b)
3 12
3 12
4 16
No real solution
36
6
c)
3
16
3
4
d)
e)
25 xy 8
3
64
3
525 x5 y11
3
( 4)3
3
53 x5 y11
5x4 y3
5 xy 3 3 x 2 y 2
f)
3( 3  6)
3
3 3
6
5
4 x3
5
22 (223)(2 3) x9
5
2532 x9
2x 5 9x4
12 x5
5
6x

2 5 3 4
3
2(5)  ( 3 2) 3 4
53 2  2
33 2
5

3
53 2  3 8
9 332
g)
3
3
4
16 x5 y 4
3
(2)3 2 x3 x 2 y 3 y
2 xy 3 2 x 2 y
2) Multiplying Radicals:
𝑛
3
8
3) Dividing Radicals:
𝑛
√𝑎
𝑛
𝑛
𝑛
𝑎
Rule: If √𝑎 and √𝑏 are real numbers and 𝑏 ≠ 0, then 𝑛 = √𝑏.
√𝑏
3
5
a)
5
15

5
5
81
3
{since both inside cube root, we can divide}
b)
3
3
3
c)
1
5
5
x2
5
x3
5
x5
d)
x3
5
x3

5
x3
x
x5
x3

3x 2 y
3y
x x
3y
3y
3y
x 3xy
3y
e)
3
3
3
5 3 2 y2
10 y 2
5


4 y 3 22 y 3 2 y 2
2y
81 3
 27  3
3
9
Practice:
Mulitply:
1) 3 6 y 4 3 4 y 5
2)
3 x 3 10 x 2 2 3 25 x8
3
24 y 9
6 x 3 (2 5)(52 ) x10
3
8 3( y 3 )3
6 x 3 2 53 x 9 x
2 y3 3 3
6 x(5)( x 3 ) 3 2 x
30 x 4 3 2 x
5
4 3
4
4
5
x
(2
x
x

3
6
x
)
3)
5 4 x 3 (2 x 4 x )  (5 4 x 3 )3 4 6 x 5 )
10 x 4 x 4  15 4 6 x8
10 x 2  15 x 2 4 6
Divide and Simplify:
4) 10  2
3
3
3
5)
5
4
{Both inside root, so divide}
6)
4
4
7)
xy 3
25 x 3 y
4
y2
4
3
52 x 2
4
52 x 2
4
25 x 2 y 2
5x
4
52 x 2
50 xy 3


3
3
4 2 3
4 2
5 xy
25 x y
5 x y 5 xy
4
2
3
xy 3
4
25 x3 y
3
y2
3
52 x 2
3
5 xy 2
5x
3
5x
3
5x
2
4
52 xy 3
4
10
11
7.3 Binomial Radical Expressions
Multiply each pair of conjugates.
1.
3

2 9 3 2 9

9 4  27 2  27 2  81
9(2)  81  63
Add or subtract if possible.
2.
9 32 3
3.
5 22 3
5.
14 3 xy  33 xy  113 xy
11 3
4.
3 7  73 x
Simplify. Rationalize all denominators. Assume that all variables are positive.
6.

2 2 3 4
2


34 2 34
7.



4 3 8
4 3 8

4 9  16 4(3)  16

8.
10  4 3  10 3  4 9
25  10 3  10 3  4 9
10  14 3  12 22  14 3

25  12
13
4 3 8
  32
4
6 45 y 2  4 20 y 2
6 9 5y  4 4 5y
2
9.
3

12 3 3  4 3 9  6 3 3
63 3  43 9

6 y 2  15 5 y  2 5 y  5 25
6 y  13 5 y  25
3
3
26 y 5
y  5 2 y 5 5
3
4(3) 3 3  2(2) 32  3( 2) 3 3
18 y 5  8 y 5
3
4 3 81  2 3 72  3 3 24
4 34  2 32 23  3 233
2
6(3)( y ) 5  4(2)( y ) 5
10.


2  2 3 (5  2 3)
2  12

5  12
5  2 3 (5  2 3 )
11.
3  10  5  2 
 5  2  5  2 
3 5  3 2  52 2  5 2 2
52
3 5 3 2 5 2  2 5
3
5 5 8 2
3
12
Notes for 7.4 Rational Exponents
Warm-up:
1) Recall: What is a rational number? (If you do not know this, look it up in your
book)
Any number that can be written in the form: a/b, where a and be
are both integers and b does not = 0.
2) On your calculator, find the value of each expression:
1
 
a)
1
 
1
 
b) 36 2 
25 2 
5
c) 49 2 
6
7
3) What do you think it means to raise a number to the ½ power?
Same as square root.
4)
On your calculator, find the value of each expression:
1
 
3
a)
1
 
 3
1
 
 3
8
b) 125
c) 27
2
5
3
5) What do you think it means to raise a number to the
1
power?
3
Cube root
6) Write each expression as a radical expression:
1
1
a) 5 6
b) 4 5
6
5
7) Simplify:
3
274
3
2
5
3
c) x 3
4
12
3
274  3 (33 )4  3 (312 )  3  34  81
8) Write each expression in exponential form:
a)
5
x
x2
x
1
5
b)
4
x3
x
3
4
13
This is because when you have a rational number as an exponent, the denominator is the
index for the radical. Also, the numerator is the power it is raised to.
Example 1:
4
3
27 is the same as 3 27 4 . This problem could be done two different ways.
First Case
3
Second Case
4
3 3
(33 ) 4
4
3
(3 )
4
4
4
3 3 3 3
3333
81
9)
3
3
4
3
34  81
Solve.
1
9 3
a) ( 27 x )
b)
3
1
9 3
3
3
9
3
((3) x )
2
2
((3) x )
2
((3)1 x 3 )
3
x3
1
2
1
2
1
2
32
1
2
1
5 2
(2 )
5
2
(2 )  2
1 5

2 2
2
6
2
 23  8
10) Now you try an example: (Can you add the fractions?)
1
2
3
4
x x x
x
x
10
20
29
20
x
15
20
1
5
{Can’t combine since different indexs/denominators}
x
4
20
 20 x 29  x 20 x9
(Remember, an exponent that is a fraction in the denominator is the same as having a
radical sign in the denominator.)
11)
Simplify:
=
x
4
5

x
=
Let’s think…. what can we multiply
the top and bottom by to get the
fraction exponent to disappear.
1
4
5
1x
4
5
1
5
x x
1
5
1
5
x
= 1 
x
5
x
x
x
4
5

1
x
4
5

15 x1
5
x4
5
x1

5
x
x
14
12)
12)
4
6
25 x
this cannot be simplified...
81
81
3
4

3
4 4
3 
12
4
3  3  27
13)
12
 32 
 32 121
x

  x
 1 
 1 12
y3 
y3 1



x
2
3
y
15)
1
3
4
12
1
4

12
1
361.5
(62 )1.5
63
or
216
8





14)
5
1
2
 243w10  5
2
10 5
3 w 
5
10
5
4
3 w
x
y
 8
4
y
x
20
5
1
1
w4
1
w4
1
1
1
1
(6 2 ) 2
w4
w4
1
63
216
3
1 4 w3
3
2 2
(6 )
9
w4
32 w4 
16)
(6 2 ) 10
3
4
4
w
4
w3
4
4
w
1w 4
w3

4
1
w3
w
3
w4 w4
4
w3
w
4
4

4
w3
w
4
3
4
3  
4
33  27
3
15
7.5:
Solving Square Root and Other Radical Equations
Warm-up: {Not in notes packet 2014}
1)
Solve: x2 = 25
2) Simplify:
5 or -5
25
5 (ONLY!)
3) Simplify:
3
125
4) Simplify:
-5
25
NO SOLUTION
Big Idea:
is looking for the PRINCIPLE Square root,
Which means, the positive root.
5) Solve:
x 5
6) Solve:
x = 25
x = 125
To UNDO a square root…. SQUARE both sides
To UNDO a cube root…
CUBE
both sides.
3
x 5
16
Practice 7-5
Solving Square Root and Other Radical Equations
Solve. Check for extraneous solutions.
1
1. ( x  2) 3  5
2.
4
3x 3 + 5 = 53
4
3
 ( x  2) 13   5 3

  


x  2  125
3 x 3  48
3
3
 43  4
4
x

16


 
 
x  127
12
12
Check {in original equation}
127  2 
125
1
3 3
5 
1
3
1
3
4
3 x 3  5  53
3
4 4
x  (2 )
12
4
x  2  23  8
 5?
4
Check: 3(8) 3 + 5 = 53?
 5?
3(2)4 + 5 = 53?
 5 YES
48 + 5 =53 YES
x 1= x  1
3.

x 1

4.
1
5. x 2  5 = 0
2
2
  x  1
2
x + 1 = (x-1)(x-1)
x + 1 = x2 – 2x + 1
No solution
radical will always
give the principle
0 = x2 – 3x
square root, so positive.
0 = x(x-3)
x = 0 or 3
ck: x = 0
2x  1 =  3
ck: x = 3
0  1  0 1
3 1  3 1
1  1  1
42
 12 
2
 x    5
 
x  25
ck: x=25
1
(25) 2  5  0
55  0
17

6.
x7

2
  x  5
1
7. (2 x  1) 3  3
2
0  x 2  11x  18
0  ( x  9)( x  2)
27  25
16  4
9  3  3
2x  5 = 7


2
10.
 7
 x
2
  x  6
2
0  x 2  13 x  36
0  ( x  9)( x  4)
x  9 or x = 4
4 x  2  3x  4
4x  2
x +6=x
x  x 2  12 x  36
x  27

3
 13 
3
 x   1
 
x 1
x  x6
2
2 x  5  49
2 x  54
11.
1
x3  1
ck: x = 2
97  95
2x  5
2x3  2
2 x  28
x  14
x  9 or 2
2=0
1
3
x  7  x  10 x  25
9.
1
2x 3
1


3
3
 (2 x  1)    3


2 x  1  27
2
ck: x = 9
8.
 
2
4 x  2  3x  4
x2
3x  4
12.

2
 x  2 3
2
4=5
2
3
( x  2)  9
3
2
3


(
x

2)

9

  2


2
3
3
2 2
x  2  3
x23
6
2
x  2  33
x  2  27
x  29

18
13. 2 x  1  26  x
2
x 1
 
2
4( x  1)  26  x
4 x  4  26  x
3 x  30
x  10
26  x
14.

2
3
2x 4 = 16
3
4
x 8
4
3
4
 
 x   8 3
 
3
4
x  2
4
3 3

12
4
3
x2
x  16
19
7-1 Roots and Radical Expressions
Find each real-number root.
1.
5.
4
144
2.  25
12
-5
0.0081
6.
3
.3
27
0.01
3.
4.
3
0.1i
7.
3
3
0.001
0.1
27
8.
-3
0.09
0.3
{ONLY get i’s when have even index}
Simplify each radical expression.
81x 4
21.
121y10
22.
9x2
25.
5
243x5 y15
5 5 5 15
3 x y
3xy 3
23.
11y5
26.
3
( x  9)3
( x  9)
3
8g 6
2g2
27.
25( x  2)4
3
24.
5x3
28.
5( x  2)2
5( x  2)( x  2)
5( x 2  4 x  4)
5 x 2  20 x  20
If no index is written, this implies an index of :
2
125x9
3
64 x9
343
4 x3
7
20