1078 Basic Algebra questions: exponents and roots
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Edward Morey October 6, 2006
(note that the questions are numbered in groups)
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Find answers to the following expression
1. 5 2 
2. 33 
9
3.
3
64 
4.
5
5. 7  711 =
6. 75  711 =
7. 75  75 =
2 x 2 x 3  3( x 2 ) 3  x(2 x 4 )  9 x 6 
From Rational Exponent Form to Radical Form and Vice Versa:
1.
a, u
b,
1
5
 _________
6x y 
2
5 29
 ___________
c, (3 xy ) 3 5  ____________
e,
4
9u  ______________
f ,  7 (2 x) 4  ____________
g,
3
x3  y 3  ________________
Working with rational exponents:
1.
a, 93/ 2  ___________
b, ( 27) 4 / 3  _____________
c, (5 y1/ 4 )(2 y1/ 3 )  ___________
d , (2 x 3/ 4 y1/ 4 ) 4  _____________
1/ 3
 8 x1/ 2 
e,  2 / 3 
 x 
 _______________
f , 2c1/ 4 (5c 3  c 3/ 4 )  ______________
g , (7 x1/ 2  y1/ 2 )(2 x1/ 2  3 y1/ 2 )  ______________
Apply the exponential properties to simplify the expressions
y x y 
4
1.
6 3
x9
y6
  y 2 x 3   y 6 x 9 
3
 x 
 9 y 6 


3
2.
3

1
2
 x3 y6 


 9 

1
2
1
 9 2
3
 3 6  3
x y 
x 2 y3
2. Rationalize the denominator:
2 x y
x 2 y
2 x y
2 x  5 xy  2 y


x  4y
x 2 y
x 2 y
x 2 y




Rational Exponents and Radicals


1. Change each expression to radical form. Do not simplify.
3
a. x 2
1
1
b. x 3  y 3
2
2 3 5
c. (25u v )
2. Change each expression to rational exponent form. Do not simplify.
a. 3 4 x3
m2  n2
b.
c.
3
(4 x 2 y ) 2
3. Simplify each expression using properties of radicals. All variables represent positive
real numbers.
a.
b.
5
( x  y )5
4
16 x 3
4
2x
c. 3 125 x 2 3 5 x 2 3 25 x 2
d.
24
x6
4. Rationalize the denominators and simplify.
3
a.
2x
x2
b.
x2
2a  2b
c.
a b
5(m  n) 2
d.
m n
5. Rationalize the numerators and simplify.
a
x a
x a
b.
x 1
x3
c.
xh  x
h
For questions 8-11 perform the indicated operation. Express your final answer as a
single, fully reduced, simple fraction with positive exponents. Be sure to list any
restrictions on the variables that need to be imposed to make your answer equivalent to
the starting expression.
x3  x2 3y 

1)
 
2 y 2  y 4 4 x 5 
x3  x2 3 y  x3  3  x3 4 x3 y 3 2 x6 y







2 y 2  y 4 4 x 5  2 y 2  4 x 3 y 3  2 y 2
3
3
2)
z4
z
 2
2
4 z  8z z  4
z4
z
z4
z
 2


2
4 z  8 z z  4 4 z  z  2   z  2  z  2 
 z  4  z  2  
z  4z
4 z  z  2  z  2  4 z  z  2  z  2 
 z  4  z  2   4 z 2  z 2  6 z  8  4 z 2  3z 2  6 z  8

4 z  z  2  z  2 
4 z  z  2  z  2  4 z  z  2  z  2 

3)
x
2
3
y x
x
2
y 3 x 4
  x
4 2
 q3
4) 
4
 16 p
 q3 
 16 p 4 



1
2

2
2




y
2

3 2
 y3 
y6
 2  4
x
x 
1
2
 q3 p 4 


 16 

1
2
1
1
 16  2
4
4q 2
 3 4  3
, or 2 2
q p
q p 
q 2 p2
5) Rationalize the denominator:

2 a 2 b
a b
 a  b   2a  4 ab  2b
ab
 a  b 
2 a 2 b
2 a 2 b

a b
a b

1. Perform the indicated operation. Express your final answer as a single, fully reduced,
simple fraction with positive exponents. Be sure to list any restrictions on the
variables that need to be imposed to make your answer equivalent to the starting
expression.
x
A
2
y 3 x 7

3
y9
x 15
Rational exponents
1. Simplify:
x2 y

5
3
1
2
( x y 1 ) 2
A xy
1
3
2. Write in the simplest radical form.
3
8x 4 y 6
A 2 xy 2  3 x
3. Evaluate the following numerical expressions
(8)
2
3
____________
A4
4. Perform the indicated operation. Express your final answer as a single, fully reduced,
simple fraction with positive exponents. Be sure to list any restrictions on the
variables that need to be imposed to make your answer equivalent to the starting
expression.
 q
  4
9p
A




1
2
3
1
2
q p2
5. Rationalize the denominator:
A

2 a 2 b
a b
 a  b   2a  4 ab  2b
ab
 a  b 
2 a 2 b
2 a 2 b

a b
a b

Exponentials
Combine into a single fraction and reduce it to the lowest terms
y 7 2y
 
6 10 15
y  21
A
30
1.
1
1
 2
x  4 x  4x  4
2x
A
( x  2)( x  2) 2
2.
2
3. Perform the indicated operation. Express your final answer as a single, fully reduced,
simple fraction with positive exponents. Be sure to list any restrictions on the
variables that need to be imposed to make your answer equivalent to the starting
expression.
x7  x2 3y 

a)
 
2 y 2  y 4 8 x 3 
z4
z
 2
b)
2
4 z  8z z  4
4x8 y
3
A a)
b)
z4
z
z4
z
 2


2
4 z  8 z z  4 4 z  z  2   z  2  z  2 
 z  4  z  2  
z  4z
4 z  z  2  z  2  4 z  z  2  z  2 
 z  4  z  2   4 z 2  z 2  6 z  8  4 z 2  3z 2  6 z  8

4 z  z  2  z  2 
4 z  z  2  z  2  4 z  z  2  z  2 

4. Combine into a single fraction:
*You do not need to expand the final expression.
1
1

2
x 9 x3
A
1
1
1
1
1
x3
x2






x  9 x  3 ( x  3)( x  3) x  3 ( x  3)( x  3) ( x  3)( x  3) ( x  3)( x  3)
2
x  2 z 3  y 1 z 
  3 
5. Simplify the following and express it only with positive exponents:
y
 x 
x 2 z 3
A 
y
2
 x3 
x 2 z 3 x 6
z3 x6 y 2
 1  

 x 4 yz
2 2
2
2
y
y
z
y
z
x
y
z


6. Simplify the expression and graph the result on [-3,3]:
2 2 x 1  2 1 2 2 x 11 2 2 x
y
 3 x  3 x  2 2 x 3 x  2  x
3x
2
2
2
2
y
8
6
4
2
x
-3
-2
-1
1. (5 x 2 y 3 )2  52 x 4 y 6 
1
2
3
y6
; distribution the –2 inside to all the factors
25 x 4
8 x 3 y 1
4 y4
4 y3


6 x 2 y 4 3x 2 x3 y 3x5
1 x
1 x 1 x
1 x
x


 (1  x) 
 (1  x)
x
3.
1
1
1

x
x 1
x
1

x
1
x
x
5
2
5
7x  x
7x
x2 7 1


 
4.
4 x5
4 x5 4 x5 4 4 x3
3x 4  4 x 2  1 3x 4 4 x 2
1
3x 1 1
 3 3 3
 
5.
3
4x
4x 4x 4x
4 x 4 x3
2.
6.
18x8 y5 z 2   9  2
7.
42 xy
6x
6x


;
7y
7y
7y
8.
18m3n4 2m3n2  36m6 n6  6m3n3 ;
9.
1/ 2
x8/ 2 y5/ 2 z 2/ 2  3 2 x4 y 2 yz  3x4 y 2 z 2 y
5 x
5 x (3  2 x )
15 x  10 x


9  4x
3  2 x (3  2 x )(3  2 x )
4( x  3)4 ( x  3)2
1
10.


;
2
4
8( x  3)
2( x  3)
2( x  3)2
11.
1 1 c2  b2
(c  b)(c  b)

2 2
b 2  c 2 b2 c 2
b2c 2

 b3 c 3 
3
3
1 1
c b
(c  b)(c 2  bc  b 2 )
b c

b3 c 3
b3c 3
b 3c 3
(c  b)(c  b)
b 3c 3
bc(c  b)


 2
2 2
2
2
bc
(c  b)(c  bc  b ) (c  bc  b2 )
12.
2x 5
2*5* x
1
2



2
20 x
4*5* x
2
13.
t  x ( t  x )( t  x )
(t  x)
1



tx
(t  x)( t  x )
(t  x)( t  x ) ( t  x )
1. x 4 / 5 x 2 / 5  x 4 / 52 / 5  x 2 / 5
3
6 1/ 3
2. (8 x y )
 4 x 2 
3.  4 
 y 
3
1
3
8 x y
1/ 3
1/ 2
6
1
3
 2 xy 2 
1/ 2
 x2 y 4 


 4 
1/ 3
 8 x 4 y 3 
4. 
2 3 
 27 x y 
xy 2
2

1/ 3
 8 y3 y3 

2 4 
 27 x x 
7
( x3  y 3 )7
b,
3
8 y3
c,
3
16 x 4 y
2 xy
Rationalize Denominator:
1.
1/ 3
 8 y6 

6 
 27 x 
Applying Properties of Radicals:
1.
a,
2x
y2

2 y2
3x 2
a,
b,
x2  y 2
x y
23 y
7 y
1. Evaluate
56  56  2  56
but not equals 106
56  56  512
54  56  54  56
54  56  510
3 1

4 4
3
8
4
35
4
1. 3x (4 x  2 x )  12 x  6 x  12 x  6 x
2. (3u1/ 2  v1/ 2 )(u1/ 2  4v1/ 2 )  3u  12u1/ 2v1/ 2  u1/ 2v1/ 2  4v  3u  13u1/ 2v1/ 2  4v
3. (5m1/ 2  n1/ 2 )(5m1/ 2  n1/ 2 )  25m  n ;
3/ 4
1/ 4
8
1. Simplify the following expression.
x 2 y
x 7  ( y 3 ) 1
A x5 y 4
2. Simplify and express in the simplest radical form:
9x3
5
3x 5 x
A
5
3. Write in scientific notation.
0.0000859
A 8.59  10 5
4. Answer True or False by writing the whole word as ‘T’ and ‘F’ can look alike:
 z 3 y 2   z 2 ( z )(  y 2 )
AT
____________