LEARN MATH STEP BY STEP THROUGH VERY EASY PROCESS

POWERS

__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 xⁿ. **

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

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

**C) (xᵐ)ⁿ = x ᵐⁿ, and ****(a.xᵐ)ⁿ = a****ⁿ. ****x 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 . xⁿ = x ᵐ⁺ⁿ**

**1) 3² X 3⁴ = 3²⁺⁴
= 3⁶ = 3 X 3 X 3 X 3 X 3 X 3 = 729**

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

**3) 5ᶟ X 5⁴ = 5ᶟ⁺⁴ = 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 – xᵐ ÷ xⁿ = x ᵐ⁻ⁿ**

**1) 3⁷ ÷ 3⁴ = 3⁷⁻⁴ = 3ᶟ
= 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)ᶟ = (4 X 4)ᶟ = (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)ᶟ = ( 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

**Example – 1) 4² = 4
X 4 = 16 = 4² (Square of 4 or 4 ^{th} square)**

** 2) 5² = 5 X 5 = 25 = 5² (Square of 5 or 5 ^{th} square)**

** 3) 7² = 7 X 7 = 49 = 7² (Square of 7 or 7 ^{th} square)**

** 4) 9² = 9 X 9 = 81 = 9² (Square of 9 or 9 ^{th} 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

**Example – 1) 4ᶟ = 4
X 4 X 4 = 64 = 4ᶟ (Cube of 4 or 4 ^{th} cube)**

** 2) 5ᶟ = 5 X 5 X 5 = 125 = 5ᶟ
(Cube of 5 or 5 ^{th} Cube)**

** 3) 7ᶟ = 7 X 7 X 7 = 343 = 7ᶟ
(Cube of 7 or 7 ^{th} Cube)**

** 4) 9ᶟ = 9 X 9 X 9 = 729 = 9ᶟ
(Cube of 9 or 9 ^{th} Cube)**