ALGEBRA - solved problems
Properties of the real numbers
 Properties Examples 1 a + b = b + a, 5 + 3 = 3 + 5, 2 (a + b) + c = a + (b + c), (2 + 3) + 5 = 2 + (3 + 5), 3 a + 0 = a, 3 + 0 = 3, 4 a + (-a) = 0, 4 + (- 4) = 0, 5 a · (b + c) = a · b + a · c, 2 · (4 + 7) = 2 · 4 + 2 · 7, 6 a · b = b · a, 6 · 8 = 8 · 6 7 a + (-b) = a - b, 9 + (- 4) = 9 - 4 = 5, 8 - (a + b) = - a - b, - (3 + 4) = -3 - 4 = -7, 9 b - a = - (a - b), 5 - 7 = - (7 - 5) = -2, 10 a · (-b) = - a · b, 3 · (- 5) = - 3 · 5 = -15, 11 (-a) · (-b) = a · b, (- 3) · (- 6) = 3 · 6 = 18, 12 - (-a) = a, - (- 7) = 7 13 a · 0 = 0, (- 11) · 0 = 0, 14 15 a · 1 = 1 · a = a, - 5 · 1 = 1 · (- 5) = - 5, 16 (-1) · a = - a, (-1) · 4 = - 4, 17 18 19 20 21 22 23 24 25 26 27 28 if since 29 if since
Properties of exponents
 Properties Examples 1) a · a · · · a = an,  n factors (exponent), a -base, a3 = a · a · a,      24 = 2 · 2 · 2 · 2 = 16, 2) a = a1,  or   a1 = a, 2 = 21,    (-3)1 = -3, 3) a0 = 1,   an ¸ an = 1, 23 ¸ 23  = 23 - 3  = 20  = 1,     (ab3)0 = 1, 4) 1n  = 1, 15 = 1,      (-1)5 = -1, 5) (- a)2n = a2n,    2n -even exponents, (- 5)4 = 54 = 625, 6) (- a)2n -1 = - a2n -1,   2n-1 -odd exponents, (- 2)3 = - 23  = - 8, 7) (-1)2n = 1, (-1)6 = 1, 8) (-1)2n -1 = -1, (-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)
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