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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 - 3  = 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 · an  = am + n, a)  a3 · a4 = a3 + 4,                 b)  24 · 25 = 24 + 5 = 29 = 512, 2 am ¸ an  = am - n, a)  x5 ¸ x3  = x5 - 3  = x2,       b)  0.14 ¸ 0.13 = 0.14 - 3 = 0.1, 3 an · bn  = (a · b)n, a)  24 · 34 = (2 · 3)4 = 64,        b)  45 · 0.85 = (4 · 0.8)5 = 3.25, 4 an ¸ bn  = (a ¸ b)n, a)  x6 ¸ y6  = (x ¸ y)6,             b)  125 ¸ 35 = (12 ¸ 3)5 = 45, 5 6 (am)n  = am · n, a)  (a3)4 = a3 · 4 = a12,            b)  (45)3 = 45 · 3 = 415, 7 8 9
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 1 2 3 4 5 6 7 8
Rationalizing denominator
 1 Example 2 Example 3 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) : c   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 : 2   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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