Unit 3
Radical Exponents and factoring
1.2: Exponents and Radicals
1.3: Factoring
1.4: Fractional Expressions
(1.2) Exponents and Radicals
A) Simplify each expression.
1) (-3x2)4
81x8
2) (-2yz3)5
3) (503x100y90)0
-32y5z15
1 (anything raised to a power of
zero is defined to be 1)
{ Reminder: When raising a power to a power, MULTIPLY exponents. }
B) Simplify
4) (3x)(5xy)
5) (2x3y4)(-5xy5z)
15x2y
6)
3x5 y
9 xy 2
x4
3y
-10x4y9z
{ Reminder: When multiplying like variables,
{ Reminder: When dividing like variables,
ADD
the exponents. }
SUBTRACT
the exponents }
C) Re-write without any negative exponents. Don’t worry about simplifying or distributing out any
exponents.
{ I think about negative exponents as not being happy living where they are, so they move, and become
happy… weird, but if it helps… }
7) 5x-3
8) 4m-1n2
9) ( 4mn2 )-3
( alot of unhappiness here )
5
x3
4n 2
m
1
(4mn 2 )3
1
64m3 n 6
D) Review Square Roots and Radicals: Solve for the possible values of each variable.
10) x2 = 16
11) y2 = 49
y 7
x 4
{This means x = 4 or x = -4}
{You are taking the SQUARE
ROOT
of both sides }
12) w = 16
{Don’t forget the
13) z =
w=4

}
49
z=7
{ Notice the difference between the first 2 (#10,11) and the last 2(#12,13). The first 2 are to find the square
root. The last 2 are asking for the radical, which is always positive. It is called the principal square root, which
means the positive one. }
E) Re-write in radical form… Don’t worry about rationalizing or simplifying yet.
1
14) x 2
2
x
15) x
x

1
2
( Note: is x happy living in the
{Normally, we don’t include the index of ‘2’}
x3
3
16) x 2
1

x
2
numerator? )
3
17) x 4
4
x3
F) Re-write in exponential form…
18)
3
x
1
x3
19)
5
x2
2
x5
20)
4
5x
1
(5x) 4
1
x
G) Simplify as much as possible.
x9
21)
22)
x2 x2 x2 x2 x
3
x7
{Now, want groups of 3}
2  2  2  2  2 xxxxx
xxxx x
x
4
2  2 xx 2 x
x
3
x x x  xx x  x
OR another method:
9
2
x x
4
1
2
1
2
x x x
4
32x5
23)
3 3
3
23
x
OR another method:
4
x
7
3
x x
2
1
3
 x2 3 x
4 x2 2 x
OR another method
5
2
2 x
2
2
2
1
2
5
2
x
2
2
1
2
1
2
2 x 2 x
1
2
4 x2 2 x
Open the TI-Nspire document Equivalent_or_Not.tns.
What does it mean for expressions to be equivalent? This
activity investigates expressions that are equivalent under
certain conditions on the variable.
Press / ¢ and / ¡ to navigate
Move to page 1.2.
through the lesson.
1. Find the value for each expression when
a. x= 2.
x2
= _____
x
x 2 = _____
|x| = _____
b. x= 4.
x2
= _____
x
x 2 = _____
|x| = _____
2. Based on your answers from question 1, predict the value for each expression when x = 15.
x2
= _____
x
x 2 = _____
|x| = _____
3. Find the value for each expression when
a. x = 3.
x2
= _____
x
x 2 = _____
|x| = _____
b. x = 8.
x2
= _____
x
x 2 = _____
|x| = _____
4. Based on your answers from question 3, predict the value for each expression when x = 20.
x2
= _____
x
x 2 = _____
|x| = _____
5. Find the value for each expression when x = 0.
a.
b.
c.
x2
x
x2
|x|
Two algebraic expressions that are equal for every substituted value of the variable chosen
from a set of numbers are said to be equivalent for that set of numbers.
6. a. Is the expression
b. Is the expression
c. Is the expression
x2
equivalent to x for the set of positive real numbers? Why or why not?
x
x2
equivalent to x for the set of negative real numbers? Why or why not?
x
x2
equivalent to x for the set of real numbers? Why or why not?
x
7. Tom says that the expression
x 2 is equivalent to x for the set of real numbers. Do you agree? Why
or why not?
8. For what values of x are
Is 9. Is the expression
3
x
x 2 and |x| equivalent? Explain your reasoning.
equivalent to x for the set of real numbers? Why or why not?
(1.2) Radicals Practice with Absolute Values
80 x5 y 2 z  245x4 xy 2 z  22 x2 y 5xz  4 x2 y 5xz
27)
x2 | y3 | z 4 x
x5 y 6 z 8
28)
From beginning, odd # of x’s x2 and z4 have even exponents
so will be positive, absolute value is not needed.
29) 6
b | c | d e2 f 2 g 2 6 a 4bc 2d 3 fg 2
a4b7c8d 9e12 f 13 g14
b = has odd exp. left inside.
d= has odd exp. left inside.
c= had even exp. originally, index is even and took out an odd # of no c’s
e, f and g = took out an even # of each
3 5 14 11
2 w h
30)
3
w2 3 w
w7  3 w6 w
31)
3
5
3
32w14 h11
14
3
2 w h
1
{Odd index, no absolute
2
3
4
11
3
2
3
2 w h
value needed}
2 w4 h 3
3
3
2
3
2 2 w2 h 2
2w4h3 3 4w2h2
32)
4
w7
3
4
w w
33)
4
4
{odd # left inside, so no
32c 6 d 10 e9 f 5
25 c 6 d 10 e9 f 5
5
4
absolute value needed}
6
4
10
4
9
4
2 c d e f
2 | c | d 2e2 f
34)
5
4 6 7
160a b c
{odd index…}
5
5
4 6 7
2 5a b c
2bc
5
5a 4bc 2
35)
1
2 2
 4x 
5
4
4
2c 2 d 2ef
4x2  2 | x |
The good/bad news is that the text says that we will assume all variables are positive, so you won’t need to
worry about absolute values on your assignments – I hate that….
36)
1
2
 6a b  5a b  
 3 6 a 2 b   51 a 7 b 
 6a 2 b  5a 7 b 
 
 

3 2 
2 
3 2 
2 
3 2 
2 


15
a
b
2
b
15
a
b
2
b
15
a
b
2
b





3
 1


2
7
 3a 9 b2 
a6
a3
  3 4

2
3
a
b
b
b


3
4 2
37)
5k 
39)
 27 y 
6
 5k
3 12

1
3

 5k
1
1
6 3
3 y 
3

6
5
1
3y2
1
6 3
38)
 27 y 
40)
2 a
8a   
 3y2
2

3
9

3
2
9 3
6
3

2 a
18
3
2 2 a 6
1
4a 6
41)
 2 r s 

2
12
 3 r s  
–1 0 2
1
4

2 
 3 r 
1
24 12 4
  2  r s
12
2

(3 r
(23 r 6 s 3 )
3

r
12
1
24 12 4
r s
1
2

1
2 4
1
2
)
8r 6 s 3 3
r 8r 24 s12 3r


 8r 5 s 3
r
r
r
3r
Simplify. Do not assume positive variables.
( x  4) 4
42)
( x  4)
43)
4
(m  3)12
(m  3)
4
2
44)
4
12
4
w
8
4
w2
| (m  3)3 |
( x  4)2 or x 2  8x  16
w8
{Even index, and odd # taken out}
3
45)
54 xy 5
3
2y
27
3
46)
54 x
y4
y
( x)
5
x
2 y
3
3
1
3
3 x y
3
4
x 7 y12 z 8
7
4
7
2
12
4
x y z
1
2
1
8
4
3
4
x y3 z 2
x | y3 | z 2 4 x3
x3 x
27 xy 4
3
47)
( x)7
4
3
3 y 3 xy
48)
3
x8 y 6 z 3
=
8
3
6
3
3
3
2
2
3
x y z  x y 2 z1  x 2 y 2 z
3
x2
Reminder: When do you have to worry about absolute values? ( Circle...)
Even or Odd index? ( Little number inside crook or radical )
Even or Odd exponent inside radical after simplifying?
Even or Odd exponent outside of radical?
(1.2) Rationalizing Denominators
The Basics… Rationalize the denominator.
5
3
1)
5 3
5 3

3
3 3
Now on to more fun stuff….
4)
8
3
( 2)
3
3
2
7)
7
5 x
7
1
5x
4 6
3
5 3
3
6)
3
x2 3 x
7 3x
5x2
3
4x x
x
4x
4 x
4 3 x2
7x
7x
x x
3
x x
3
x2
x
10 y 3 y
3
x
4x
3
x
4x
3
x
y 3 2 5y
3
6
x
x
3
2
3
10 3
3
4x
x
4x
x
3
5
3
7
5x
5
4
10
x
y
3
7x 2
53 x 5
7
3
3
10 3
2 3 3
8 6
3
6
3
125x5
3
4
10
12
10
223
10 xy 4
8)
7 x2
7
3
3)
8 6
6 6
5)
( 2)3
3
8
6
2)
I think it helps to
simplify before
rationalizing.
3
3
77x x
77x x
y 3 490 x 2 y
7x
(1.3) Polynomials
Degree of Polynomial are based on the
highest degree term within the polynomial.
{Degree of term is found by adding the exponents on the variables}
Examples: Find the degree of each polynomial.
2) 5x4 + 7x
1) 7
{this is like 7xo}
Degree = 0
4
3) 12x3y2 – 7xy3
1
Degree = 4
5
4
Degree = 5
Polynomial answers are written in descending order. ( Highest degree 1st )
{ If the same degree, look at alphabetical order next }
Write each polynomial in descending order. (Also called Standard Form)
4) 4x – 8x2 + 7
5) 5xy + 8x2 – 7y3 + 6
-8x2 + 4x + 7
-7y3 + 8x2 + 5xy + 6
Notice: 8x2 and 5xy each have degree = 2.
Alphabetically, x comes before y, so take
larger # of x’s 1st.
6) 7x8 – 3x7y + 9x6y2 – 4x5y3 + 10y6
Already in standard form…. descending order.
1st 4 terms have degree = 8, so we go from descending degree of x…
Adding/ Subtracting Polynomials: { Add/Subtract only like degree terms. Write in descending order. }
{ I circle or underline like terms when there are lots of terms }
7) ( 4x2 + 5xy + 3y2 ) + ( 3x2 – 12xy + 4y2 )= 7x2 – 7xy + 7y2
7x2 – 7xy + 7y2
8)
( 4x2 + 5xy + 3y2 ) - ( 3x2 – 12xy + 4y2 )
{The subtraction between quantities, mean that we need to
having a ( ) + -1( )}
distribute out the negative. It is like
4x2 + 5xy + 3y2 -3x2 + 12xy - 4y2 = 1x2 + 17xy - 1y2
=
1x2 + 17xy - 1y2
Which of the following terms are polynomials?
9) xy
10) 3x + 7y
Polynomial
Polynomial
11)
4x
- 5x + 6y
y2
12) 7x - 4 x - 8
NOT
NOT
(variable in denominator)
(radical variable)
{is like a negative exp. on
{is like a fractional
variable}
exponent on variable}
A. Multiply:
(x+5)(x+5)
{ Hint : re-write without exponents }
x(x+5) + 5(x+5) ( Remember to FOIL or double distribute….
x(x+5) + 5(x+5)
The answer is NOT x + 25 )
x2 + 5x + 5x + 25
13) ( x + 5 )2 =
2
x2 + 10x + 25
14)
( 4x2 + 5xy + 3y2 ) ( 3x3 – 12xy + 4y2 )
4x2 ( 3x3 – 12xy + 4y2 )+ 5xy( 3x3 – 12xy + 4y2 ) + 3y2 ( 3x3 – 12xy + 4y2 )
12x5 – 48x3y +16x2 y2 + 15x4y - 60x2y2 + 20xy3 + 9x3y2 – 36xy3 + 12 y4
12x5 + 15x4y + 9x3y2 – 48x3y - 44x2y2– 16xy3 + 12 y4
12x5 + 15x4y + 9x3y2 – 48x3y - 44x2y2– 16xy3 + 12 y4
15)
( w -3k)( w + 3k)
w ( w + 3k)  3k( w + 3k)
w+3k w  3k w  9k 2
w  9k 2
9k 2  w
16)
( 4x – w )3
(4x-w)(4x-w)(4x-w)
(4x-w)(16x2 -4wx -4wx +w2)
(4x-w)(16x2 -8wx +w2)
4x(16x2 -8wx +w2) -w(16x2 -8wx +w2)
64x3 -32wx2 +4w2x -16wx2 +8w2x –w3
64x3 -48wx2 +12w2x – w3
-w3 + 12w2x -48wx2 + 64x3
17) ( x + 2 )( x – 4 ) ( x + 3 )
(x+2) ( x2 –x – 12 )
x( x2 –x – 12 ) + 2( x2 –x – 12 )
x3 – x2 – 12x + 2x2 – 2x - 24
x3 + x2 – 14x – 24 = x3 + x2 – 14x - 24
B. Factor:
9) 7x3 + 21x2
7x2(x+3)
11) y2 – 9
(y-3)(y+3)
13) 8k2 – 200
8( k2 – 25)
8( k – 5 ) ( k + 5 )
Simplify:
14)
12 x3 w2  18 xw4
8 xw2
10) w2y3 – 25wy2
wy2(wy - 25)
12) m2 – 36
(m - 6)(m + 6 )
Multiply:
15)
16)
17)
(x - y) (x2 +xy + y2 )
( x – 5 ) ( x2 + 5x + 25 )
( y + 3 ) ( y2 – 3y + 9 )
18) (3x + 2y ) ( 9x2 –6xy + 4y2 )
F. Factor
19) 25w3 – 8
21)
m2 + 5m + 6
20)
22)
8k3 + 27m3
a2 + 3a – 10
23) 12x2 – 8x - 15
24) 50x2 + 45xy – 18y2
25) 5x3 +10x2 - 20x – 40
26) yx +2y – 3x – 6
27) x2 – 10x + 25 – 16y2
Group Discussion:
What is the best way to factor each of the following expressions? Does it matter if you factor difference of
squares or difference of cubes first?
28) x6 - 26
30) y6 – 5y3 + 4
29) y6 – 27z12
31) x12 – 1
(1.4) Introduction to Fractional Expressions
Factor:
1) x2 – 4
2) x2 + 4x + 4
(x+2)(x-2)
(x+2)(x+2)
4) y2 – 9
5) z2 – 12z + 27
(y-3)(y+3)
3) y3 – 27
(z - 9 )(z – 3 )
(y-3)(y2 +3y + 9)
6) 4z2 – 35z – 9
(4z +1 )(z – 9)
Simplify each fractional expression.
7)
( x  2) ( x  2)
x2  4

x 2  4 x  4 ( x  2) ( x  2)
8)
( x  2)
( x  2)
{you go… we go… can’t cancel!!!}
9)
z 2  12 z  27
4 z 2  35 z  9
( z  3) ( z  9)
(4 z  1) ( z  9)
( z  3)
(4 z  1)
2
y 3  27 ( y  3) ( y  3 y  9)

y2  9
( y  3) ( y  3)
( y 2  3 y  9)
( y  3)
10)
3 x 2  3 xy  7 x  7 y
9 x 2  49
3 x( x  y )  7( x  y )
(3 x  7)(3 x  7)
( x  y ) (3 x  7)
(3 x  7) (3 x  7)
( x  y)
(3 x  7)
Adding/Subtacting/Multiplying and Dividing Fractional Expressions
Perform the indicated operation, then simplify as much as possible.
11)
4
7

25 30
12)
4 y
7 x

x 2 y xy x
4y
7x

x2 y x2 y
4 y  7x
x2 y
14)
y2  4
y6
 2
2
y  4 y  12 y  36
Divide means multiply by reciprocal
Find LCD: (5)(5) and 5(6)
Need 2 5’s and a 6 = 150
46
75

25 6 30 5
24 35
59


150 150 150
13)
( x  4)
x 2  9 x  18

x 2  7 x  6 x 2  x  12
1
( x  4)
( x  6)1 ( x  1)
1
x 1
15)
1

( x  6) 1 ( x  3)
( x  4)1 ( x  3)1
y2  4
y 2  36

y 2  4 y  12 y  6
1
( y  2) ( y  2)
( y  6)1 ( y  2)1
y2
25 x 2  4 15 x 2  19 x  10

x2  2 x  1
3x 2  8 x  5
25 x 2  4 3 x 2  8 x  5
x 2  2 x  1 15 x 2  19 x  10
{Do easy factor 1st, then use to give hints}
(5 x  2)(5 x  2) (3x  5)( x  1)
( x  1)( x  1) (5 x  2)(3 x  5)
5x  2
x 1
16)
1

( y  6) 1 ( y  6)
y  61
4
5

w3 w
4w
5 ( w  3)

w( w  3) w( w  3)
4 w  5w  15
w( w  3)
9 w  15
w( w  3)
9 w  15
w2  3w
17)
4
5

w3 w 2
4( w  2)
5( w  3)

( w  3)( w  2) ( w  2)( w  3)
4 w  8  5w  15
( w  3)( w  2)
9w  7
w2  w  6
{I see that 9w-7 doesn't contain
a factor of the denominator,so I
multiply it out right away.}
18)
4w
3w
5


w2  9 w2  5w  6 w2  w  6
4w
3w
5


( w  3)( w  3) ( w  2)( w  3) ( w  3)( w  2)
4 w( w  2)
3w( w  3)
5( w  3)


( w  3)( w  3)( w  2) ( w  2)( w  3)( w  3) ( w  3)( w  2)( w  3)
4 w  8w  3w2  9 w  5w  15
( w  3)( w  3)( w  2)
2
w  22 w  15
{check if factor of (x+3) goes in...}
( w  3)( w  3)( w  2)
2
w  22 w  15
( w2  9)( w  2)
2
w  22 w  15
3
w  2 w2  9 w  18
2
Fast-food workers:
19) A survey of high school students found that ½ of the female students had jobs and
2
/3 of the male students had jobs. It was also found that ½ of the female students, who
worked, worked in fast-food restaurants and 1/6 of the male students who worked,
worked in fast-food restaurants. If equal numbers of male and female students were
surveyed, then what fraction of the working students worked in fast-food restaurants?
Simplify:
x y
2x  y
20)
x y
2x  y
1 1
( x  y )(  )
x y
21)
1 1
( x  y )(  )
x y
22) At a Trigonometry University, ¼ of the undergraduate students commute, and 1/3 of the graduate students
commute. One-tenth of the undergraduate students drive more than 40 miles daily, and 1/6 of the graduate
students drive more than 40 miles daily. If there are twice as many undergraduate students as there are
graduate students, then what fraction of the commuters drive more than 40 miles daily.
23)
16 x 2  y 2
2 x y
{This next problem is a stretch, and won’t be on an assessment, but is interesting}
1
1
(1  x ) (2 x)  (2 x  3)   (1  x 2 ) 2 (2 x)
2
1
2 2
24)
1
2 2 2
[(1  x ) ]
Rationalizing the numerator.
25)
Rationalize the numerator.
x h
x
Express as a polynomial. (Book says: Express as a sum in the form axr )
x2  4 x  6
26)
x4
x2  4 x  6
27)
x