Indices and Surds

Indices

1. If x^N is given then x is the base and N is the index or power or exponent
2. A³ means A multiplied with itself 3 times i.e. A x A x A
3. A ^p X a ^q = a ^(p+q)
4. A ^p / A ^q = A ^(p-q)
5. (A ^p)^q = A ^pq
6. A ^–p = 1 / A^p
7. p√a = a ^1/p i.e. pth root of a
8. (ab)^p=a^p x b^p
9. A⁰ = 1 (provided a ≠0)
10.  A¹ = a
11. _A mⁿ = a ^p, where p = mⁿ ie. A raised to the base m raised to the power
12. If a ^p = b ^p, then if p is ≠0, then a = b, if p is odd and [a = b or a = (-b) if p is even]
13. If a ^p = a ^q and a ≠ 0 or -1, then p = q
14. A ^-1 = 1/A
15. (A/B)^-1 = B/A
16.  (A)^m/n = (n√A)^m
17.  √A X √B = √(AB)

Surds

1. They are irrational numbers
2. When an irrational number is simplified, the remainder which cannot be simplified and is normally expressed in the form of square root is called a surd.
3. Normally for exam questions, number whose square root cannot be further found out as a perfect rational number are surds. 4 is not a surd, as square-root of 4 is 2, where 2 is a surd as square root of 2 is 1.414… which is not a rational number.
4. To solve simplification problems regarding surds, square the numbers
5. For 1 / (p + √q) or 1 / (p + √q + √r) kind of problems, to simplify, multiply by the conjugate, which is (p – √q) or (- p + √q) for 1^st case and (p + √q – √r) or (p – √q + √r) for the 2^nd case
6.  = .555555 hence, whenever in a decimal form there is a repeated number; a dot is mentioned over it.
7. Rationalising Surds:
   When you have a fraction where both the nominator and denominator are surds, rationalising the surd is the process of getting rid of the surd on the denominator. To rationalise a surd you multiply top and bottom by fraction that equals one. Take the example shown below
   1/√2
   To rationalise this multiply by effectively 1
   1/√2 * √2 /√2
   Can you see why √2 /√2 was chosen? This is because √2 * √2 = 2 so the denominator becomes surd free.
   
   For a more complex term
   
   
   
   
   Rationalizing the surd now