When a certain number a is taken n times and multiplied in succession ( n – 1 ) times, the continued product so obtained is called the nth power of a and is written in short as aⁿ ,
Also n is called the index of aⁿ , and a is called the base of aⁿ
Therefore , a × a = a² ; ( the square )
a × a × a = a^ 3 ( the cube )
Now , 1×1×1×1×1×1×……… up to n 1’s = 1.
i . e. 1ⁿ = 1
and , oⁿ = 0
Fundamental Index law :
Hence if m and n are positive integer
( i ) a ^ m × a^ n = a ^ ( m+n ),
( ii ) a ^ m / a ^ n = a ^ ( m- n) , ( m > n )
( iii) (a^m )ⁿ = a ^ ( mn )
( iv) ( a b )ⁿ = aⁿ bⁿ
Roots of a number :
( i ) If x and y are two real numbers such that y² = x then x is called the square root ( 0r second root ) of y and is denoted by ±a ^ ( 1/ 2 ) or ±√ a .For example since 4² = 16 and
( - 4 )² = 16 the square root of 16 are 4 and - 4 .
( ii 0 For two real number a and b if b ³ = a , then b is called the cube root of a and b and is written as b = a (⅓)
( iii ) similarly , if two real numbers x and k be such that xⁿ = k , where n is a positive integer , then x is called the nth root of k ,and is written in short x is called the nth root of k , and is written in short as ; x = k^ ( 1/n) For example 2 = ( 32 )^ 1/5 , since
(2 ) ^5 = 32 ,
Some Deductions :
( i ) a^ 0 = 1 , ( ii ) a^ ( -m ) = ( 1 / a )^ m
( iii ) ( ( a ^ m ) ^n )^p = a ^ ( mnp)
(iv ) ( a / b )ⁿ = aⁿ bⁿ
( i )For real numbers a, b if a ^ x= b^ y, ( a≠ 0, 1 , ±∞ ) , then x = y ,
From aⁿ = bⁿ , we have a ^( x – y ) =a ^ 0 . and ( x – y ) = 0 or ( x = y )
( ii ) If a ^ x = b ^ x , then a = b or x =0 if a≠ b then a ^ x = b ^ x , we have
( a/b ) ^ x = ( a / b ) ^ 0 . x = 0.
Prob : 1
Find the values of the given quantity
( ( 16 ) ³) ¼ = ( 16 )¾ = ( 2^ 4 ) ¾ = 2 ³ = 8
Prob: 2
Simplify = (√(( a^8)^ √( a ^ 6.√ (a ^ ( -4 )) )^ (1/ 5 )
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