Improving minhashing : De Bruijn sequences and primitive roots for counting trailing zeroes
Math for 800 07 powers, roots and sequences
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- Powers - Roots - Sequences
Transcript of Math for 800 07 powers, roots and sequences
CONTENTS
POWERS
1
2
3
...n
n times
a a
a a a
a a a a
a a a a a
EXPONENTS
Square
Cube
0 1, 0a when a
ZERO EXPONENT
0
0
0
2 1
5 1
11
4
when:
, and n is even
0na
0a
0a
PROPERTIES OF THE
EXPONENTS
4
3
4
2 16
2 8
2 16
when:
, and n is odd
0na
0a
PROPERTIES OF THE
EXPONENTS
3
3
2 8
3 27
even
odd
positive positive
positive positive
even
odd
negative positive
negative negative
ODD/EVEN EXPONENTS
4
5
2 16
2 32
4
5
2 16
2 32
, when n > 0
is undefined
0n 0 0n
00
POWERS OF ZERO
2m m ma a a
ADDITION OF POWERS
3 3 32a a a
2 2 23 5 8a a a
mm ma b a b
If a 0, b 0, and m 1, then
22 2a b a b
0m ma a
SUBTRACTION OF
POWERS
3 3 0a a
2 2 27 4 3a a a
mm ma b a b
If a 0, b 0, a b, and m 1, then
33 3a b a b
m n m na a a
MULTIPLICATION OF
POWERS
3 4 72 2 2
72 2 2 2 2 2 2 2
74
3
22
2
DIVISION OF POWERS
mm n
n
aa
a
42 2 2 2 2 2 22 2 2 2 2
2 2 2
1k ka a a 1k
k aa
a
MULTIPLICATION/DIVISION
OF POWERS
n
m m na a
POWERS TO A POWER
4
3 122 2
4
12
2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
2
m m mab a b
POWER OF A PRODUCTS
2 2 22 3 2 3
2
2 2
2 3 2 3 2 3
2 3
p
m n mp npa b a b
2
4 5 8 102 3 2 3
2
4 5 4 5 4 5
4 5 4 5
8 10
2 3 2 3 2 3
2 3 2 3
2 3
2 2
2
3 3
4 4
POWER OF QUOTIENTS
m m
m
a a
b b
2 2
2
3 3 3 3
4 4 4 4
na
1 1n
n
na
a a
1 n
na
a
NEGATIVE EXPONENTS
3
3
12
2
3
3
12
2
NEGATIVE EXPONENTS
1n
na
a
POWERS
Edwin Lapuerta, May 2014
ROOTS
3
...n
n times
b a a a b
c a a a a c
d a a a a a d
ROOTS
Square root
Cubic root
a b a b
a b a b
ADDITION/SUSTRACTION
OF ROOTS
9 16 9 16
25 16 25 16
a b ab
ab a b
MULTIPLYING ROOTS
4 9 36 6
36 4 9 4 9 6
a a
bb
a a
b b
DIVIDING ROOTS
36 362
99
36 362
9 9
a a b
b b b
a b
b
RATIONALIZATION
2 2 3
3 3 3
2 3
3
1
mn mn
nn
a a
a a
FRACTIONAL EXPONENT
23 23
1122
8 8 4
9 9 9 3
1
1
n
n
n n
a a
a a
If a ≥ 0
FRACTIONAL EXPONENT
31
3
13 3
2 2
2 2
n
n
n n
a a
a a
If a ≥ 0
FRACTIONAL EXPONENT
3
3
3 3
2 2
2 2
mm n n
m nn m
a a
a a
If a ≥ 0
ROOT OF A ROOT
22 3 3 6
32 3 62
64 64 64 2
64 64 64 2
ROOTS
SOLVING EXPONENTIAL
EQUATIONS
If ax = ay, then x = y ( a ≠ 0 and a ≠ 1).
EXPONENTIAL
EQUATIONS
SEQUENCES
SEQUENCE
The first term of a sequence is
represented by a1, the second term
a2, and so on to the nth term, an.
2 1 1 2..., , , , , ,...n n n n na a a a a
SEQUENCE
2 3 4 5 6..., , , , , , ...a a a a a
ARITHMETIC SEQUENCES
A sequence in which each term, after the
first, if found by adding a constant, called
the common difference, to the previous
term.
2, 5, 8, 11, 14, …
2
5
8
11
14
a1 a2 a3 a4 a5
2, 5, 8, 11, 14, …
1 1 1 1 1
1 2 3 4
, , 2 , 3 ,..., 1
, , , ,..., n
a a d a d a d a n d
a a a a a
2, 5, 8, 11, 14, …
1 1na a n d
GEOMETRIC SEQUENCES
A sequence in which each term after the
first is found by multiplying the previous term
by a constant called the common ratio.
2, 6, 18, 54, 162, …
a1 a2 a3 a4 a5
2, 6, 18, 54, 162, …
2 3 1
1 1 1 1 1
1 3 3 4
, , , ,...,
, , , ,...,
n
n
a a r a r a r a r
a a a a a
2, 6, 18, 54, 162, …
1
1
n
na a r
Apple Inc. Cash GrowthBullish Cross 2013 Outlook
(in millions)
ARITHMETIC
AND
GEOMETRIC
SEQUENCES
SEQUENCES
SUMMARY
