Laws of indices

Laws of indices

Before discussing about the laws of indices, we will discuss about base, index and root.

Base and index:

If m is an integer,

  aᵐ = a×a×a×...×a(m times).

  Here, ''a" is called base and "m" is called index of "a".
   That is, in aᵐ , the number "a" itself is called base and how many times we multiply it is called it's index or power.

   As for example,

    2⁵= 2×2×2×2×2

    (-3)⁵= (-3)(-3)(-3)(-3)(-3)

     x⁴= x×x×x×x

     Here, 2, -3, x are called base and 4,5 are index.

    Root:

If a and x be two real numbers and n be any positive integer  such that, aⁿ=x , then, a is called as n-th root of x and denoted by , ⁿ√x  or (x)ˡ/ⁿ.

 In particular, if n=2,3 then a is called as the square and cube roots of x respectively.
example, Let, a²= 64, a=?
 Here, we have, 64=8².
Now, a² = 8²
       or, a=+8 or -8.
Here 8 is the square root of 64.

Note:

For, square root of a number 25(say) , we have two results +5 and-5.

For, cube root of a number, one and only one is positive.

In general, for n-th root  of a number one and only one positive root.

Laws of indices

(1) (aᵐ)×(aⁿ)=aᵐ⁺ⁿ
(2) (aᵐ)÷(aⁿ)=aᵐ⁻ⁿ
(3) (aᵐ)ⁿ= aᵐⁿ
(4) (ab)ᵐ= (aᵐ)×(bᵐ)
(5)(a÷b)ᵐ= (aᵐ)÷(bᵐ)
(6) If m is a positive number, a⁻ᵐ=1÷(aᵐ). Here, a⁻ᵐ is called reciprocal of aᵐ.
(7) If m,n are integer, aᵐ/ⁿ means (aᵐ)ˡ/ⁿ ; i.e., n-th root of aᵐ.
(8) If m=0, a⁰ is meaningless, a⁰=1.
(9) If a,m,n real and aᵐ=aⁿ then, m=n, where, a not equal to 0,1,-1.
(10) If a, b, m are real , and aᵐ= bᵐ then, either a=b or m=0.

   Some examples:

  (1) calculate, (2⁵)×(2⁻³) =?

    Ans:  we have, (2⁵)×(2⁻³) = 2⁵⁻³ = 2² = 4.

  (2) calculate, (8²)÷ (2³) =?

    Ans:  we have, 8²= (2³)² = 2⁶.

   Now,(8²)÷(2³) =(2⁶)÷(2³) = 2³=8.

  (3) simplify,( 2⁵)× (5⁵)=?

Ans: we have, (2⁵)×(5⁵) = (2×5)⁵ = 10⁵.

  (4) simplify, (9⁴)÷ (3⁴)=?

  Ans: We have, (9⁴)÷(3⁴)= (9÷3)⁴ =3⁴.

  (5) calculate, {(⁵√8)⁵/²} ×{(16)⁻³/⁸ }=?
 Ans: we have, (⁵√8)⁵/² =(8)¹/² =(2³)¹/²=2³/².
           Also, (16)⁻³/⁸ =(2⁴)⁻³/⁸ = 2⁻³/².
    so,{(⁵√8)⁵/²} ×{(16)⁻³/⁸ }=2³/² × 2⁻³/² =2⁰=1.

  (6) Arrange the following numbers in increasing order: 2⁶³ , 3⁴⁵ , 5²⁷ , 6¹⁸ .

  Here, 2⁶³ = (2⁷)⁹ = (128)⁹ ;
             3⁴⁵ =(3⁵)⁹ = (243)⁹ ;
              5²⁷ =(5³)⁹ = (125)⁹ ;
              6¹⁸ = (6²)⁹ = (36)⁹ ;
  Since, 36<125<128<243
   so, 6¹⁸<5²⁷<2⁶³<3⁴⁵.
 Therefore, the increasing order is :
                       6¹⁸, 5²⁷, 2⁶³ ,3⁴⁵.
                 
        

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