Basics of Exponents

Exponents - Basics of Exponents

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When a unavoidable whole 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 .

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Exponents

Therefore , a × a = a² ; ( the quadrilateral )

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 unavoidable 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 whole :

( i ) If x and y are two real numbers such that y² = x then x is called the quadrilateral 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 quadrilateral root of 16 are 4 and - 4 .

( ii 0 For two real whole 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 unavoidable 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 - When a unavoidable whole 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 quadrilateral )

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 unavoidable 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 whole :

( i ) If x and y are two real numbers such that y² = x then x is called the quadrilateral 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 quadrilateral root of 16 are 4 and - 4 .

( ii 0 For two real whole 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 unavoidable 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 .
c 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 .

I hope you obtain new knowledge about Exponents . Where you may offer easy use in your day-to-day life. And most significantly, your reaction is passed about Exponents .