INDICES
If x be any real number and n is the natural number, then
xn = x C x C x C………. C x upto n factors. and n is index of x plural of index in indics.
Laws of Indices :
- xm C xn = xm+n; x Î R+, m, n, ÎN
- xm ÷ xn = xm-n; x ÎR+; m, nm ÎN
- (xm)n = xmn; x ÎR+; m, n, Î N
- (xy)m = xmym; xy Î R, m Î M
- (x + y)m = xm + ym = x, y Î R+; m Î M
- x0 = 1; where x is non-zero real 1
- x-n =— , x Î R, n a non-negative.
xn
- If x and y are two positive real numbers and n a natural number, such tha yn = x, then y = x1/n.
- If x is a positive real number and m and n are integers with n positive,
1 m
then—— = x-mn
xn
- If x, y are positive real numbers and n a natural number, then
nÖx C nÖy = nÖx C y.
- If xm = ym, then x = y
- If xm = xn, then m =
Pure Surd : It is that surd which has unity as its rational factor, the other factor being irrational Ö5,5Ö5 are pure surds.
Mixed Surd : It is that surd which has a rational factor other than unity, the other
factor being irrational is called a mixed surd e.g.
3Ö5, 43Ö4 etc.
Similar Surds : Surds having the same irrational factor are known as similar
surds or like surds, e.g.
5Ö3 and 7Ö3.
Rationalisation of Surds
Monomial Surd : This surd is one which contains a single term e.g.
Ö3, Ö5, 3Ö75 etc.
Binomial Surd : It is one which contains two monomial surds or a monomial
surd and a rational number e.g.
Ö3 + Ö5, Ö7 + Ö9 etc.
Trinomial Surd : It is one which consists of three terms at least two of which are monomial surds e.g.
Ö3 + Ö7 – Ö8; 5 + Ö6 etc.
Rationalising Factor : When the product of two surds is a rational number each is known as rationalising factor (R.F.) of the other, e.g. Öx + Öy is the rationalising factor of Öx – Öy because the product is x – y which is a rational number.
Quadratic Surd : Surds of quadratic order are quadratic surds.
or Binomial surds are called quadratic surds.
Sequence of operations in an Arithmetical Expression : If no parentheses is present in an expression then the various arithmetical operations are carried out in the following sequence.
Powers
Extraction of roots Multiplication Division
Additions and Subtractions.
Example
3.4 + 6 ÷ 2 + 52 – Ö16 + 1 = 12 + 3 + 25 – 4 + 1 = 37
However if the expression contains a parentheses then follow procedure given above starting from the innermost parentheses (in sequence : round brackets, square brackets, double brackets). After elemination of brackets follow the procedure given above.
Example
3 + 5 C {4 C 3 – [5 C 4 + (2 C 3 – 3 + 1) – 20] + 12}
= 3 + 5 C {4 C 3 – [5 C 4 + (6 – 3 + 1) – 20] + 12}
= 3 + 5 {4 C 3 – [5 C 4 + 3 + 1 – 20] + 12}
= 3 + 5 {4 C 3 – [20 + 3 + 1 – 20] + 12}
= 3 + 5 {4 C 3 – 3 + 1 + 12}
= 3 + 5 {12 – 3 + 1 + 12}
= 3 + 5 C 22
= 3 + 5 C 22
= 3 + 110
= 113
Prime Numbers : Any natural number (excluding 0 and 1) that can be divided only by 1 and by itself is known as a prime number. e.g. 3,5,7,11,13,17 etc.
