1
Algebra practice part 4
2
E. Exponents
3
Positive exponents
Examples:
43  4  4  4  64
41  4
4 0  1 (convention)
3-rd power of 4, 4: base, 3: exponent
In general:
xn  
x  
x  x 
... x
n factors
(x any number,
n positive integer)
Exercises:
(3) 4  (3)  (3)  (3)  (3)  81
 
 34   34   3  3  3  3  81
(3)5  (3)  (3)  (3)  (3)  (3)  243
x 0  1 if x  0
4
Negative exponents
Examples:
1
1
1 1
3
1
4  3
4  1
4 64
4 4
In general:
1
(x any non-zero number,
n
x  n
n positive integer)
x
1
x-1 is the inverse of x
x 
x
Exercises:
1
1
1 5
2

  
2 2
5
5
1
x
y
  
x
 y
5
Radicals
Example:
?3 = 8
• 23=8: 2 is the 3-rd root (cubic root) of 8
• the 3-rd root of 8 is denoted by
i.e.
3
8 2
3
8
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Radicals
Example:
?3 = –8
• (–2)3=8: –2 is the 3-rd root of –8
• the 3-rd root of 8 is denoted by
i.e.
3
 8  2
3
8
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Radicals
Example:
?4 = 16
• 24=16: 2 is a 4-th root of 16
• (–2)4=16: also –2 is a 4-th root of 16
• 16 has two 4-th roots: 2 and -2
• positive 4-th root of 16 is denoted by
4
16
i.e. 4 16  2
• it follows that the negative 4-th root of 16 is
given by 
4
16
i.e.  4 16  2
8
Radicals
Example:
?4 = –16
• no numbers whose 4-th power equals –16
• –16 has no 4-th root
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Radicals
• 16 has two 4-th roots:
4
16  2 and  4 16  2
this is a typical example of the case of an even root of a
positive number
• –16 has no 4-th roots
this is a typical example of the case of an even root of a
negative number
• 8 has one 3-rd root:
3
8 2
this is a typical example of the case of an odd root of a
positive number
• –8 has one 3-rd root:
3
 8  2
this is a typical example of the case of an odd root of a
negative number
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Radicals: remarks
• 3-rd roots are cubic roots
2
• 2-nd roots are square roots:
• for any positive integer n: n
9  9 3
0 0
n
1 1
• in many cases roots have to be calculated using
the calculator:
♦
♦ …
2  1.414...
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Fractional-exponent-notation for roots
Example:
2  4 16
1
 16 4
23 8
1
 83
In general:
n
x
1
xn
(x any stricly positive number,
n positive integer)
Exercises:
3 23
4 0 .2 
1
45
1
 32
 3   1.732...
0.5
 5 4  1.3195...
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More general fractions as exponent
Examples:
2
83
8
8

3
stands for
2
3
1.5
2
8 3
8

3
2
 8
3

1
3
82
2

82 , i.e. 3 82  3 64  4
1
1


64
4
3
1
83

1
512
 0.044...
In general:
z
xn
 x
n
z
(x any strictly positive number,
z integer, n positive integer)
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Irrational exponents

4 4
3.141 592 653 589 793 238 462 643 383 279 502 884 197 1...
 43.141 592 653 589 8
 77.8802336...
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Product of powers with same base
Example:
x3  x4 can be written in a simpler form :
x  x  (
x 
x
 x)  ( 
x
 x
 x
 x)  x

3 factors
4 factors

3
4
3 4
x
7
3 4 factors
In general (real exponents and positive bases):
xr  x s  xr s
Exercise:
1
1
3

2

1
1
1
2
2
2
2
x  2  x x  x
x  3 
3
x
x
x2
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Quotient of powers with same base
Example:
x5 / x3 can be written in a simpler form :
5
x

3
x
x 
x 
x 
x
x

5 factors
x 
x
x


x  
x 
x 
x
x

5 factors
x 
x 
 x

3 factors
3 factors
 x 53  x 2
In general (real exponents and positive bases):
xr
r s

x
xs
Exercise:
x
x
2

1
x2
x
2

1
2
x2
x

3
2

1
3
x2

1
x3
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Power of a power
Example:
(x3)2 can be written in a simpler form :
x 
3 2

x 3

x3  
x 
x
 x
x 
x
 x  x 32  x 6
2 factors
3 factors
3 factors




2 factors
In general (real exponents and positive bases):
x 
r s
Exercise:
1 5
1
5



5
5
4
x   x 4   x 4  x 4
 
 
x
r s
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Power of a power: a special case
 x
2
?
 1
x2


 
x2 
1
x2 2
x2
x  2
x2  x
2
1

2
  x 2  x1  x


rational exponents for
x
2
1
2
positive bases only, not
valid for x= –2
 x1  x
!
x 2  22  4  2  x
!
x 2  (2) 2  4  2  x
ONLY for positive x-values!
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Product of powers with same exponent
Power of a product
Example:
x3y3 can be written in a different form:
x3  y 3  (
x 
x
 x)  ( y  y  y )  ( x  y )  ( x  y )  ( x  y )  ( x  y ) 3
 
3 factors
3 factors
3 factors
(xy)3 can be written in a different form
In general (real exponents and positive bases):
x  y 
r
Exercise:
x  y  (x 
1
y) 2

1
x2
 xr  y r

1
y2
 x y
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Quotient of powers with same exponent
Power of a quotient
Example:
x3/y3 can be written in a different form:
x
x x x x x x  x 

     
3
y y y y y y  y
y
3
3
In general (real exponents and positive bases):
r
x
xr
   r
y
 y
Exercise:
1
1
x
x  x 2 x2
    1 
y  y
y
2
y
20
Sum of powers with same exponent
Power of a sum
Examples:
x  y 3  x3  y 3
x  y 2 x 2  y 2
=
=
x 2  2 xy  y 2
( x  y )  ( x  y) 2
=
( x  y)  ( x 2  2xy  y 2 )
=
x3  3x 2 y  3xy 2  y 3
In general:
(x+y)r can NOT be written in a simpler form:
x  y 
r
 xr  y r
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Sum of powers with same exponent
Power of a sum
In general:
x  y 
r
x y
r
r
x  y 
r
 xr  y r
Further examples:
x y

x
||
x  y 
y
||
1
2

1
x2

1
y2
1
x y
||
x  y 1

1 1

x y
||

x 1  y 1
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Rules for exponents: summary
for all real exponents and positive bases:
same base:
power of a power:
same exponent:
xr
r s

x
xs
xr  x s  xr s
x 
r s
x
r s
x  y r  x r  y r
r
x  y   x r  y r
applied to (square) roots:
x
x

x y  x  y
y
y
r
x
xr
   r
y
 y
x y  x  y
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Equations with powers: example 1
The volume of a cube with side x is given by V=x3.
1. Find the volume of a cube having side 4 cm.
2. What is the side of a cube having volume 729 cm3?
3. A first cube has side 3 cm. Find the side of a
second cube, whose volume is the double of the
volume of the first one.
Answers:
1. 64 cm3
2. solving x3=729 gives x=7291/3=9 (cm)
3. solving x3=233 gives x=321/3=3.77…3.8 (cm)
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Equations with powers: example 2
Write y in terms of x if y3 = 5x2.
we have to get rid
of the exponent 3
( y3 )1/3=(5x2 )1/3
y = 51/3(x2)1/3
Answer: y = 51/3x2/3
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E. Exponents
Handbook
Chapter 0: Review of Algebra
0.3 Exponents and Radicals
(except: rationalizing denominators, i.e. example
3, example 6.c, problems 59-68)