10. Indices.

 

10. Indices.

Learn and Remember :

1. Indices : If a is a rational number and m is a natural number, then the product

a × a × a ... m    times is written as a^m

i.e. a × a × a... m times = a^m 

In a^m, a is called the base and m is called the index or the exponent.

2. Laws of indices :

(i) Rule for the product of index numbers when the numbers have the same base :

If a is any rational number and m and n are any two positive integers, then

1) am  × an = a m +n

^{e.g.  34  × 32 = 34+2 = 36}

(ii) Rule for the quotient of index numbers when the numbers have the same base :

If a is any non-zero rational number and m and n are positive integers such that

m > n , then

2)   am ÷ an  = am-n

e.g.5⁹ ÷ 5³= 5^9-3  =5⁶

 If m<n, then

3) am ÷ an   = 1 ⁄ an-m

e.g.

7^{2  ÷} 7⁵   =¹ ⁄ _{7^{5-2  =1 ⁄ 73}}

(iii) Rule for the index of the exponent :

If a is any rational number and m and n are positive integers, then

4) (am)n   =  am × n

e.g.  (2³⁾⁴ =2^{3 ×4} = 2¹²

(iv) Rule for the index of the product of the numbers :

If a and b are any rational numbers and m is any positive integer, then

5) (a × b)m = am × am

(v) Rule for the index of a number in the ^a ⁄ _b form :

 If  ^a ⁄ _b  is any rational number and m is any positive integer, then

6)   (a ⁄ b)m = am ⁄ b m

(vi) Rule for determining the value of a non-zero rational number when the index is zero :

If a is any non-zero rational number, then

7)     a° = 1

(vii) Rule for converting a negative index into a positive index :

(1) If a is a non-zero rational number and m is any positive integer, then

8)   a-m = 1 ⁄ am

e.g. 6^-5 = ¹ ⁄ _{6 ⁵}

(2) a and a^-1   are multiplicative inverses of each other, because a ×a^-1  = a × ¹ ⁄ _a= 1.

In general, for a non-zero number m, _a^m  ×_a^-m = 1 ,as  _{a^{m  and a-m   are}} multiplicative inverses of each other.

e.g. 11^{3 ×} 11^-3 = 11³ ×¹ ⁄ _{11^{3  = 1}}

∴  11³  and 11^-3 are multiplicative inverses of each other.

3. Expanded form of numbers:

We have 1 = 10°, 10 = 10^{1 ,}  100^{2 ,  etc.}

etc. ¹ ⁄ ₁₀ = 10^-1    ,¹ ⁄ ₁₀₀ = 10^-2

  It becomes easier to express very large numbers as well as very small numbers by expressing these numbers in the exponential form with base 10.

eg . 1) the distance between Saturn and Uranus is 14,39,00,00,00.000 m. Using powers of 10, this distance in a short form is 1.439 × 10¹² m.

(11) the diameter of the atom of oxygen is 0.00 00 00 00 00 000356 mm. This we can write in short form using powers of 10 as 3.56 × 10 ^-14   mm.