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ALGEBRA
  Exponentiation - powers or indices, rules and properties
Exponentiation properties Examples
 1   a a a = an,       n factors 1  a3 = a a a,      24 = 2 2 2 2 = 16
2  a = a1,  or   a1 = a 2  2 = 21,      (-3)1 = -3
3  a0 = 1,   an  an = 1 3  23 23  = 23 -= 20  = 1,     (ab3)0 = 1
4  1n  = 1 4  15 = 1,      (-1)5 = -1
5  (- a)2n = a2n,    2n -even exponent 5  (- 5)4 = 54 = 625
6  (- a)2n -1 = - a2n -1,   2n-1 -odd exponent 6  (- 2)3 = - 23  = - 8
7  (-1)2n = 1 7  (-1)6 = 1
8  (-1)2n -1 = -1 8  (-1)7 = -1
  The rules for powers (or exponents)
Rules Examples
 1   am a = am + n,   a)  a3 a4 = a3 + 4,                 b)  24 25 = 24 + 5 = 29 = 512,
2  am a = am - n,   a)  x5 x3  = x5 - 3  = x2,       b)  0.14 0.13 = 0.14 - 3 = 0.1,
3  an b = (a b)n,   a)  24 34 = (2 3)4 = 64,        b)  45 0.85 = (4 0.8)5 = 3.25,
4  an b = (a b)n,   a)  x6 y6  = (x y)6,             b)  125 35 = (12 3)5 = 45,
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6  (am) = am n,   a)  (a3)4 = a3 4 = a12,            b)  (45)3 = 45 3 = 415,
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  Radicals - Exponentiation with a fractional exponent
Properties Examples
  If m, n and p are natural numbers (n >1) and if  a and b are nonnegative real numbers, then
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  Rationalizing denominator
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Example  
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Example  
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Example   
  Algebraic expressions - monomial, binomial, trinomial, . . . , polynomial
  Expanding algebraic expression by removing parentheses or brackets
 The square of binomial - perfect square trinomial Example
   
 The cube of binomial Example
   
 The binomial expansion algorithm
 
Example
 
  Factoring algebraic expressions
  - difference of two squares
  - difference of two cubes
  - sum of two cubes
 
 
 
 
 
 
  Ratios and proportions
     If     a : b = k    and    c : d = k,
then     a : b = c : d   and    a d = b c,   and    (a + b) : b = (c + d) : d,
  (a + b) : a = (c + d) : c,    (a - b) : a = (c - d) :  and   (a + b) : (a - b) = (c + d) : (c - d).
     If     a : b = b : c     then   is the geometric mean
     If     a1 : b1 = k,    a2 : b2 = k,    a3 : b3 = k,   . . .   ,  an : bn = k
then,     a1 : a2 : a3 :  . . .  : an = b1 : b2 : b3 :  . . .  : bn .
If given are proportions:    a : b = i : j,     b : c = k : l   and   c : d = m : n
                         then,     a : b = i : j = ( i k m) : ( j k m),
                                     b : c = k : l = ( j k m) : ( m j l)
                           and     c : d = m : n = (m j l) : ( n j l),
            follows     a : b : c : d = ( i k m) : ( j k m) : ( m j l) : ( n j l).
  Example
   Given,    a : b = 1 :   and   b : c = 3 : 4
    then,     a : b = 1 : 2 = (1 3) : (2 3),
                b : c = 3 : 4 = (3 2) : (4 2),    therefore    a : b : c = 3 : 6 : 8.
  Percentage
Denoting, the base or whole value (x), rate (p), part of amount (y) then,
    part : whole = rate : 100     or     y : x =  p : 100     that is
  Examples
 a)  What is 5% of 30?   From the proportion,
 b)  What is the base value if 15% of it is 30?
 c)  What percent of 30 is 27?
 
 
 
 
 
 
 
 
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