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1
Algebra practice part 4
2
E. Exponents
3
Positive exponents
(convention)
(x any number,n positive integer)
3-rd power of 4, 4: base, 3: exponent
Examples:
In general:
Exercises:
4
Negative exponents
(x any non-zero number,n positive integer)
x-1 is the inverse of x
Examples:
In general:
Exercises:
5
Radicals
?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.
Example:
6
Radicals
?3 = –8
• (–2)3=8: –2 is the 3-rd root of –8
• the 3-rd root of 8 is denoted by
i.e.
Example:
7
Radicals
?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
i.e.
• it follows that the negative 4-th root of 16 is
given by
i.e.
Example:
8
Radicals
?4 = –16
• no numbers whose 4-th power equals –16
• –16 has no 4-th root
Example:
9
Radicals
• 16 has two 4-th roots: and
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:
this is a typical example of the case of an odd root of a positive number
• –8 has one 3-rd root:
this is a typical example of the case of an odd root of a negative number
10
Radicals: remarks
• 3-rd roots are cubic roots
• 2-nd roots are square roots:
• for any positive integer n:
• in many cases roots have to be calculated usingthe calculator:
♦
♦ …
11
Fractional-exponent-notation for roots
(x any stricly positive number,n positive integer)
In general:
Exercises:
Example:
12
More general fractions as exponent
(x any strictly positive number,z integer, n positive integer)
Examples:
stands for , i.e.
In general:
13
Irrational exponents14
Product of powers with same base
x3 ⋅ x4 can be written in a simpler form :
In general (real exponents and positive bases):
Example:
Exercise:
15
Quotient of powers with same base
x5 / x3 can be written in a simpler form :
In general (real exponents and positive bases):
Example:
Exercise:
16
Power of a power
(x3)2 can be written in a simpler form :
In general (real exponents and positive bases):
Example:
Exercise:
17
Power of a power: a special case
ONLY for positive x-values!
rational exponents forpositive bases only, not
valid for x= –2
18 Product of powers with same exponentPower of a product
x3⋅y3 can be written in a different form:
(x⋅y)3 can be written in a different form
Example:
Exercise:
In general (real exponents and positive bases):
19Quotient of powers with same exponentPower of a quotient
x3/y3 can be written in a different form:Example:
Exercise:
In general (real exponents and positive bases):
20 Sum of powers with same exponentPower of a sum
(x+y)r can NOT be written in a simpler form:
Examples:
In general:
=
==
=
21 Sum of powers with same exponentPower of a sum
In general:
Further examples:
22
Rules for exponents: summary
for all real exponents and positive bases:
same base:
same exponent:
power of a power:
applied to (square) roots:
23
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)
24
Write y in terms of x if y3 = 5⋅x2.
y3 = 5⋅x2
y = 51/3⋅(x2)1/3
Answer: y = 51/3⋅x2/3
( )1/3 ( )1/3
Equations with powers: example 2
we have to get rid of the exponent 3
25
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)