Transcript 3-rd root
1
Algebra practice part 4
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E. Exponents
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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
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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
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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
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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
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• 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
23 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 53 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 32 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
x2
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:
x3y3 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
(xy)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
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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=233 gives x=321/3=3.77…3.8 (cm)
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Equations with powers: example 2
Write y in terms of x if y3 = 5x2.
we have to get rid
of the exponent 3
( y3 )1/3=(5x2 )1/3
y = 51/3(x2)1/3
Answer: y = 51/3x2/3
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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)