Indices

• aⁿ is defined as a×a×a×... n times, where a is any real number and n is natural number. Here a is called the base and n is called index or exponent or power. The word indices is plural of index.
• By convention:
  a¹ = a
  a^-n = 1/aⁿ, a 0
  a^m.aⁿ = a^m+n
  a^m/n = a^m-n, a 0
  (a^m)ⁿ = a^mn
  (ab)^m = a^m.b^m
  , b 0
  a⁰ = 1, a 0
• You also learnt how to simplify/evaluate algebraic expressions using the above rules. You also learnt how to express algebraic expressions using positive indices only.

Exercise

1. Calculate the value of
   (i) 5³ (ii) 3⁵      (iii) (3²)³   (iv) 3^-1 +3⁰+3¹
   (v)          (vi)
   (vii) 8^-2×2⁷
2. Simplify the following:
   (i) (6g⁵h²)/(3gh)       (ii)
   (iii) e^-2/e⁶           (iv) 2g² (g³ -g  +1/g -1/g³).
3. Express the following using positive indices only:
   (i) 3 x² y^-3 z^-1      (ii) u^-1 + v^-1 = f^-1
   (iii) (xy-1)^-2
4. Simplify
   (i) y^{a -2}. y^{2 -a} (ii) (2x)³ (x^-1)⁰
   (iii) (a^-1 + b^-1)/(ab)^-1
5. Prove that (a +b)^-1 (a^-1 +b^-1) = 1/(ab)

Answers

1. (i) 125      (ii) 243     (iii) 729     (iv) 13/3
    (v) 7         (vi) 4/5     (vii) 2
2. (i) 2 g⁴h    (ii) -8y⁶/(27x³y⁶)      (iii) e^-8
    (iv) 2g⁵ -2g³ +2g -2/g
3. (i) 3x²/(y³z)               (ii) 1/u + 1/v = 1/f
   (iii) y²a⁴c²
4. (i) 1            (ii) 8x/9               (iii) a +b