1
7.1 Roots and Radical Expressions
Warm-up:
I. Exponents and Roots
1. Find the values of x n for different values of x and n.
x 1
x2
x2
(1)2=
(2)2=
x3
(1)3=
(2)3=
x4
(1)4=
(2)4=
x5
(1)5=
(2)5=
x3
(3)2=
x4
(4)2=
x5
(5)2=
2. Go back and find the nth root of each number (in the shaded rows))
Simplify:
3) (.2)2
4) (.3)2
5) (.4)2
6) (.5)2
7) (.2)3
8) (.3)3
9) (.4)3
10) (.5)3
11) (.2)4
12) (.3)4
13) (.4)4
14) (.5)4
Parts of a radical:
a
b
‘a’ is called the ___________________________
‘b’ is called the: ___________________________
x6
(6)2=
2
If no index is written, this implies an index of : ______________
II. Finding Real Roots
Find each real root.
8
15. 3
27
16
16. 4
81
17. 4 625
8x6
19. 5 243y 5
20.
1
81
21. 4 256x 4 y12
22. 0.01
23.
3
18.
24.
27.
3
4
0.0081
25.
3
27
26.
0.09
28.
5
32
29.
0.001
3
4
27
16
Summarizing:
Even Roots
Of a negative number
Of a positive number
Odd Roots
3
III. Practice with Roots and Variables.
30.
33.
36.
5
9x6 y12
31. 3 125x 6 y12
d 20
34.
n 4n
x
37.
32.
4
81a16b20

 x  4 2
35. 3 x 2  6

6
3n 12n
x
IV. What happens when the exponents aren’t multiples of the
index? (assume all variables are positive)
38.
27x5 y12
40. 4 27x5 y11
39.
3 32x 6 y8
41. 3 625x5 y11
4
Find the real-number solutions of each equation.
42. x2 = 4
43. x4 = 81
44. x2 = 0.16
45. x2 =
16
49
46. 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.
V. A twist? Simplify:
47)
50)
53)
56)
3
4
5
( 2) 2
49)
3
( 2)3
52)
3
( x )3
54)
4
( 2) 4
55)
4
( x)4
57)
5
( 2)5
58)
5
( x )5
22
48)
(2)3
51)
(2) 4
(2)5
Summary:
even
variable =
odd
( x)2
variable =
Unless the instructions say: “do NOT assume all variables are positive,” you do not need to
worry about absolute values.
5
VI. Practice:
Find all the real cube roots of each number.
59. 343
60.
0.064
61.
1000
27
Find all the real fourth roots of each number. (The book says the answer can be plus and
minus). It is wrong!
62. 81
63. 0.0001
6
7
7.2 Multiply and Divide Rational Functions
1) Simplifying Radicals (assume all variables are positive):
3
a) √50𝑥 5
b) √54𝑥 8
c)
2) Multiplying Radicals:
𝑛
𝑛
𝑛
𝑛
𝑛
Rule: If √𝑎 and √𝑏 are real numbers, then √𝑎 × √𝑏 = √𝑎𝑏.
a) √3 ∙ √12
c)
3
e)
g)
5
16
4 16
b)
4
d)
3
25 xy 8
3( 3  6)
f)
3
2 5 3 4
4 x3
3
5
12 x5
5
6x

3
5x4 y3

3
16x 5 y 4
8
3) Dividing Radicals:
𝑛
𝑛
√𝑎
𝑛
𝑛
𝑎
Rule: If √𝑎 and √𝑏 are real numbers and 𝑏 ≠ 0, then 𝑛 = √𝑏.
√𝑏
3
5
a)
c)
e)
1
5
d)
x2
3
b)
5
4y
3
81
3
3
x5
3x 2 y
9
Mulitply:
1)
3
3)
5 4 x3 (2 x 4 x  3 4 6 x5 )
6 y4
3
4 y5
Divide and Simplify:
4) 10
3
3
6)
5
4
4
xy 3
25 x 3 y
3
3
2
8
2) 3x 10 x 2 25x
5)
7)
4
4
3
3
2
25x 3 y
xy 3
25 x 3 y
10
11
7.3 Binomial Radical Expressions
Multiply each pair of conjugates.
3
1.

2 9 3 2 9

Add or subtract if possible.
2.
9 32 3
3.
5 22 3
4.
3 7  73 x
5.
143 xy  33 xy
Simplify. Rationalize all denominators. Assume that all variables are positive.
6.
2
2 34
8.
6 45 y 2  4 20 y 2

10. 3 y  5
7.
 2
y 5 5
9.

11.
2  12
5  12
4 3 81  2 3 72  3 3 24
3  10
5 2
12
13
Review 7.1-7.3
1) Simplify each radical expression.
2) Do NOT assume variables are positive. Simplify.
a)
14
15
Notes for 7.4 Rational Exponents
#1-8: Warm-up:
1) Recall: What is a rational number? (If you do not know this, look it up in your book)
2) On your calculator, find the value of each expression:
1
 
2
a)
1
 
2
1
 
2
b) 36
25
c) 49
3) What do you think it means to raise a number to the ½ power?
4)
On your calculator, find the value of each expression:
1
 
a)
1
 
1
 
b) 125 3 
8 3 
c) 27 3 
5) What do you think it means to raise a number to the
6) Write each expression as a radical expression:
1
1
2
a) 5 6
b) 4 5
c) x 3
7) Simplify:
3
274
8) Write each expression in exponential form:
a)
5
x
b)
4
x3
1
power?
3
16
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
Solve.
4
3 3
(33 ) 4
4
3
9)
Second Case
(3 )
4
4
4
3 3 3 3
3333
81
1
9 3
a) (27 x )
3
3
4
3
34  81
b)
1
2
27 * 27
1
2
10) Now you try an example: (Can you add the fractions?)
1
2
3
4
x *x *x
1
5
(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
=
Let’s think…. what can we multiply
the top and bottom by to get the
fraction exponent to disappear.
17
Simplify where possible.
12)
6
25x 4
13)
12
 32 
14)  x 
 1 
y3 


16)
18) w
361.5
1
4
15)
3
81 4
2
10 5
 243w 
17) x-1.4
18
19
7.5:
Solving Square Root and Other Radical Equations
Warm-up:
1)
Solve: x2 = 25
3) Simplify:  25
2) Simplify:
25
4) Simplify:
25
Big Idea:
is looking for the PRINCIPLE Square root,
Which means, the positive root.
5) Solve:
x 5
6) Solve:
3
x 5
To UNDO a square root…. ___________________ both sides
To UNDO a cube root… _______________________ both sides.
20
Examples: Solving Square Root and Other Radical Equations
Solve. Check for extraneous solutions.
1.
1
( x  2) 3
=5
2.
4
3x 3
+ 5 = 53
21
Flow chart to help us solve:
3.
x 1= x  1
4.
2x  1
=3
5.
1
x2
5=0
22
6.
9.
x7
=x5
2x  5 = 7
7.  2 x  1 = 3
8. 2x  2 = 0
1
3
1
3
10.
x +6=x
23
11.
4 x  2  3x  4
13. 2 x  1 
26  x
12.
14.
 x  2 3  4 = 5
2
3
4
2x = 16