# CLASS-7POWERS

POWERS

Powers For best of your knowledge, if we multiply digit with the same digit again and again then the power of that digit will be how many time we have multiplied the same digit and the number of multiplication would be the power of the said same digit, this is called Power Of Exponential of the number. The number multiplied by itself repeatedly is called the Base of the power and the number of times it is been multiplied is called the Exponent or Index. Let us assume x is Base and n is its Index

Then, that can be written as x.

Example –

1) If 4 is multiplied by 4 by 5 times then that can be written as

4 X 4 X 4 X 4 X 4 =  4⁵ =  1024

Here 4 is based and 5 is an index.

2)  If the number 10 is multiplied by 10 by 7 times then that can be written as –

10 X 10 X 10 X 10 X 10 X 10 X 10 = 10⁷ = 1,00,00,000

Here 10 is based and 7 is an index.

3) If the number 2/3 is multiplied by 2/3 by 4 times then that can be written as –

2          2          2          2                      2⁴          16

----- X ------ X ------ X ------ = ( 2/3 )⁴ = ------- = -------

3          3          3          3                      3⁴         81

Some important fact to be remembered about Powers –

If we consider any non-zero number which is x, then –

x⁰ = 1,

x¹ = x,

and    x ⁻¹  = 1 / x

In x, if x is base and n is its index then please remember the said index n can be of two types which is even or odd.

If n is odd, which is n = 1, 3, 5, 7,……… and Base is the number with a negative sign like -1, -2, -3, -4,……….

Then it can be multiplied like  = (-1)¹  = -1 ,

2)   (-1)² = (-1) X (-1) =  1, [ multiplication of even numbers of  negative ‘-‘ signs arise positive ‘+’ sign, multiplication of base would be as per the normal multiplication process ]

3)  (-1)⁵ = (-1) X (-1) X (-1) X (-1) X (-1) =  -1   [ multiplication of odd numbers of  negative ‘-‘ signs arise negative ‘-’ sign, multiplication of base would be as per the normal multiplication process ]

4)  (-2)⁴ = (-2) X (-2) X (-2) X (-2) = 16  [ multiplication of even numbers of  negative ‘-‘ signs arise positive ‘+’ sign, multiplication of base would be as per the normal multiplication process ]

5)  (-3)⁵ = (-3) X (-3) X (-3) X (-3) X (-3) = - 243 [ multiplication of odd numbers of negative ‘-‘ signs arise negative ‘-’ sign, multiplication of base would be as per the normal multiplication process ]

Some important notes about the Laws of Exponents –

A)  xᵐ . xⁿ =  x ᵐ⁺ⁿ, and xᵐ . b xⁿ = (a.b) x ᵐ⁺ⁿ = ab ᵐ⁺ⁿ, where dot ‘.’ Implies multiplication, x is Base, m & n nothing but considered as an Index.

B)  x ÷ x =  x ⁻ⁿand xᵐ ÷ b xⁿ = (a ÷ b) ⁻ⁿ, x is Base, m & n both are considered as an Index.

C)  (xᵐ)ⁿ =  x ᵐⁿ, and (a.xᵐ)ⁿ =  aⁿ. ᵐⁿ, x is Base, m & n both are considered as an Index.

Example –

A) When two powers of a number are multiplied, the product is the number raised to the sum of the exponents -   xᵐ . x  =   x ᵐ⁺ⁿ

1)  3² X 3⁴ =  3²⁺  =  3⁶  = 3 X 3 X 3 X 3 X 3 X 3 =  729

2)  4² X 4ᶟ =  4²⁺=  4⁵  = 4 X 4 X 4 X 4 X 4 = 1024

3)  5 X 5⁴ =  5⁴ =  5⁷ = 5 X 5 X 5 X 5 X 5 X 5 X 5 = 78125

B)  when a power of a number is divided by another power of the number, the result is the number raised to the power of the difference of the exponents –  x ÷ x  =   x ᵐ⁻ⁿ

1)  3⁷ ÷ 3 = 3⁷⁻ =  3  =  3 X 3 X 3 =  27

2)  5⁶ ÷ 5⁴ = 5⁶⁻⁴ = 5² = 5 X 5 = 25

3) 10⁷ ÷ 10⁴ = 10⁷⁻⁴  =  10 X 10 X 10  = 1000

C) When the power of a number is raised to a power, the result is the number raised to the product of the exponents - (xᵐ)ⁿ =  x ᵐⁿ

Example-

1) (4²) = (4 X 4) = (4 X 4) X (4 X 4) X (4 X 4) = 4⁶ = 16 X 16 X 16 = 4096

2) (5⁴)² = ( 5 X 5 X 5 X 5 ) X ( 5 X 5 X 5 X 5 ) = 5⁸ = 625 X 625 = 390625

3) (3⁴)= ( 3 X 3 X 3 X 3 ) X ( 3 X 3 X 3 X 3 ) X ( 3 X 3 X 3 X 3 ) = 3¹² =  531441

SQUARE If the same number is multiplied by two times then the index would be 2 and this is called SQUARE of a number raised the power of 2

Example1)   4² = 4 X 4 = 16 = 4² (Square of 4 or 4th square)

2)   5² = 5 X 5 = 25 = 5²  (Square of 5 or 5th square)

3)   7² = 7 X 7 = 49 = 7²  (Square of 7 or 7th square)

4)   9² = 9 X 9 = 81 = 9²  (Square of 9 or 9th square)

CUBE - If the same number is multiplied by three times then the index would be 3 and this is called CUBE of a number raised the power of 3

Example1)  4 = 4 X 4 X 4 = 64  =  4  (Cube of 4 or 4th cube)

2)  5 = 5 X 5 X 5 = 125  = 5  (Cube of 5 or 5th Cube)

3)  7 = 7 X 7 X 7 = 343  = 7  (Cube of 7 or 7th Cube)

4)  9 = 9 X 9 X 9 = 729  = 9  (Cube of 9 or 9th Cube)