Exponents & You

When a certain number a is taken n times and multiplied in succession ( n – 1 ) times , thecontinued 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×……… upto 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 = a ⁿ 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 x ⁿ = k , where n is a positive integer , then x is called the nth root of 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 = a ⁿ b ⁿ

( 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 )

= ( a ^ 8√ ( a ^ 6 . ) a ^ ( - 2) ) ^ (1/5 )

= ( a^ 8 √ a ^ 4 ) ) ^ (1/5 )

= ( a ^ 8 . a ^ 2 ) ^ ( 1/5 )

= ( a ^ (10 / 5 ))
= a ²

Prob :3

Simplify
( a^2 ( m+n ) . a ^ ( 3m – 8n ) ) / a ^ ( 5m – 7n)

= a ^ ( 2m + 2n +3m – 8n ) / a ^ ( 5m – 7n )

= a ^ ( 5m – 6n ) / a ^ ( 5m – 7n )
= a ^ ( 5m – 6n – 5m + 7n )
= a ⁿ

Simplify

(1 / ( 1 + x ^ ( b – c ) + x ^ ( c – a ) )+ ( 1 / ( 1 + x ^ ( a – b) + x ^ ( c- b ) ) + ( 1 / ( 1 + x ^ ( a - c ) + x ^ ( b - c ) )

=
x^a / x ^a( 1+ x ^ ( b- c ) + x ^ ( c – a ) ) + x ^ b / x ^b (( 1 + x ^ ( a – b) + x ^ ( c- b)) +x ^c / x ^ c (1 + x ^ ( a - c ) + x ^ ( b - c ) )

=
x ^ a /( x ^a + x ^b + x ^ c) + x ^ b /( x ^a + x ^ b + x ^ c ) + x ^ c (( x ^a + x ^ b + x ^c)

= ( x ^a + x ^b + x ^c ) / ( x ^a + x ^ b + x ^ c )

= 1

Try solving a few related problems and you will surely gain a command over the topic.

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