Mathematics Formula Algebra - Exponents And Powers

    Mathematics Formula Algebra 

      In algebra, we have various topics having formulas and identities, Some are listed below:- 

• Algebraic Identity
• Exponents and Powers
• Least Common Multiple (LCM) & Highest Common Factor(HCF)

     Exponents and Powers

      if we factorize 4,9,16 or 25 then can be written as

    4   = 2 x 2 x 1

    9   = 3 x 3 x 1

    16 = 2 x 2 x 2 x 2 x 2 x 1

    25 = 5 x 5 x 1

      The above expression can be also written in a shorter way using exponents in this way.

    4   = 2 x 2 = 2²

    9   = 3 x 3 = = 3²

   16 = 2 x 2 x 2 x 2 x 2 = 2⁴

    25 = 5 x 5 = 5²

    ^{In the above expression that represents repeated multiplication of the same factor is called power.}

    The number 2 is called the Base and 4 is known as Power or Exponents or Index.

    The conventions for reading are such as

        The above short notation stands for the product 2  x 2 x 2 x 2. Here '2' is called the base and '4' the exponent. The above number is read as " 2 raised to the power of 4" or simply as the "fourth power of 2". It is also called the exponential form of 16.

Example 1:- Express 64 as the power of 2?

Solution:- By prime factorisation method we have

     64 = 2  x 2 x 2 x 2 x 2  x 2  x 1

     So we can say 64 = 2⁶

Example 2:- Express 128 as the power of 2?

Solution:- By prime factorisation method we have

    128 = 2  x 2 x 2 x 2 x 2  x 2 x 2 x 1

     So we can say 128 = 2⁷

Example 3:- Express 81 as the power of 3?

Solution:- By prime factorisation method we have

  81 =3  x 3 x 3 x 3 x  1

     So we can say 81 = 3⁴

Example 4:- Express 320 as a product of power of prime factors?

Solution:- By prime factorisation method we have

       320 = 2  x 2 x 2 x 2 x 2  x 2  x  5 

              = 2⁶ × 5

                = 320

Laws of Exponents and Powers

The followings are important laws of exponents and are very useful to solve problems;

Laws of Exponents - 1

On Multiplying Power with the Same Base

       For any non zero real number a , where m and n are any rational numbers then 

                    a^m × aⁿ = a^{m + n}

^{or in generally we can say on multiplication if bases are equals then powers may be added.}

^{it can be easily verified by following illustrative example;}

Example 5:- Evaluate  2³ × 2² ?

Solution:- By general concepts we have

 2³ × 2²  = (2  x 2 x 2) x (2 x  2)

             = 2  x 2 x 2 x 2 x  2

            =  2⁵

     So we can say   2³ × 2²  = 2³⁺²

We have to note that the base in  2³ and   2² is same and the sum of the exponents,i.e, 3 and 2 is 5.          

Laws of exponent - 2
On Dividing Power with the Same Base

       For any non zero real number a, where m and n are any rational numbers then 

                    a^m÷ aⁿ = a^{m - n}

^{or in general, we can say on division if bases are equals then powers may be subtracted.}

^{it can be easily verified by following illustrative examples;}

Example 6:- Evaluate  2³ ÷ 2² ?

Solution:- By general concepts we have

 2³ ÷ 2²  = [ 2 x 2 x 2 ] ÷ [ 2 x  2 ]

             = 2 

            =  2¹

     So we can say   2³ ÷ 2²  = 2^{3 - 2}

We have to note that the base in  2³ and   2² is same and the difference of the exponents,i.e, 3 and 2 is 1. 

We have consider now one more illustration  for verification of above results       

Example 7:- Evaluate 10 ⁵ ÷ 10 ² ?

Solution:- By general concepts of mathematics we have,

 ^{10 5 ÷ 10 2}  =  (10  x 10 x 10 x 10 x 10 ) ÷ (10 x  10)

             = 10  x 10 x 10

            =  10³

     So we can say  ^{10 5 ÷ 10 2}  = 10 ^{5 - 2}

We have to note that the base in  10⁵ and   10² is the same and the difference of the exponents,i.e, 5 and 2 is 3.           

Laws of Exponents - 3 
On Multiplying Power with the Same Exponents

       For any non zero real number a and b, where m is any rational number then 

                    a^m × b ^m = (a x b)^m = (ab)^m 

^{or in general, we can say on multiplication if bases are different but powers are equals then bases may be multiplied.}

^{it can be easily verified by following illustrative example;}

Example 8:- Evaluate  2² × 5 ²?

Solution:- By general logical concepts we have,

 2² × 5²  = (2  x 2 ) x (5 x  5)

             = (2  x 5 ) x (2 x  5)

            = 10 x 10

            =  10²

     So we can say   2² × 5²  = 10²

We have to note that the base in  2² and   5² is different but power is the same 2 and the products of the bases,i.e, 2 and 5 are 10.          

Laws of exponent - 4
On Dividing power with the Same Exponents

       For any non zero real number a and b, where m is any rational  number then 

                    a^m÷ b ^m = (a / b) ^m

^{or in general, we can say on division if bases are different and powers are equals then bases may be divided.}

^{it can be easily verified by following illustrative examples;}

Example 9:- Evaluate  8³ ÷ 2³ ?

Solution:- By general concepts we can write,

 8³ ÷ 2³  = [ 8 x 8 x 8 ] ÷ [ 2 x  2 x 2 ]

             = 4 x 4 x 4 

            =  4³

     So we can say   8³ ÷ 2³  = ( 8 / 2) ³  =  4³

Now in the above illustration the bases in  8³ and   2³ is  different but power is same so the bases of the exponents,i.e, 8 and 2 is divided hence result is 4 so required solution is 4³. 

We may consider now one more illustration  for verification of above results       

Example 10:- Evaluate 100 ⁵ ÷ 10 ⁵ ?

Solution:- By general concepts we have

 ^{100 5 ÷ 10 5}  

=  (100  x 100 x 100 x 100 x 100 ) ÷ (10 x 10 x 10 x 10 x 10)

= 10  x 10 x 10 x 10 x 10            

=  10⁵

So we can say  ^{100 5 ÷ 10 5}  = ( 100 / 10 ) = 10 ⁵ 

Now in the  above examples   100⁵ and   10⁵  bases  is  different but power is same so the bases of the exponents,i.e, 100 and 10 is divided hence result is 10 so required solution is 10⁵.          

  

Laws of exponent - 5
On taking Numbers with Exponents ZERO

For any non zero real number a 

                    a ⁰ =  1

^{or in general, we can say that any number ( except 0 ) raised to the power ( or Exponent ) 0  is always ONE ( 1 ).}

^{It can be easily verified by following illustrative examples;}

Example 11:- Evaluate  8³ ÷ 8³ ?

Solution:- By general concepts we have

 8³ ÷ 8³  = [ 8 x 8 x 8 ] ÷ [ 8 x  8 x 8 ]

             = 1 x 1 x 1 

            =  1

By using laws of Exponents  a^m÷ aⁿ = a^{m - n}  again we can write in other way 

8³ ÷ 8³

= 8 ^{3 - 3}

=  8 ⁰

So we have  8³ ÷ 8³   =  8 ⁰   =  1 

We may consider now one more illustration  for verification of above results       

Example 12:- Evaluate 1001 ⁵ ÷ 1001 ⁵ ?

Solution:- By general concepts we can write in the following way

 ^{1001 5 ÷ 1001 5}  

= (1001  x 1001 x 1001 x 1001 x 1001 ) ÷  (1001  x 1001 x 1001 x 1001 x 1001 ) 

 = 1  x 1 x 1 x 1 x 1

= 1

By using laws of Exponents  a^m÷ aⁿ = a^{m - n}  we can write 

^{1001 5 ÷ 1001 5}  

= 1001 ^{5 - 5}

=  1001⁰

So, we have  ^{1001 5 ÷ 1001 5}   =  1001⁰  = 1 

Laws of exponent - 6
On taking Power of Power

For any non zero real number a 

                   ( a ^m ) ⁿ =  a ^{m n} = ( a ⁿ ) ^m 

^{where m and n are rational numbers.}

^{it can be easily verified by following illustrative examples;}

Example 13:- Evaluate   ^{( 2 2 ) 3} ?

Solution:- By general concepts we that it means  2 ²  is multiplied three times itself

  ( 2 ² ) ³ 

= [ 2 ²  ] x [ 2 ² ] x [ 2 ² ]             

= 4 x 4 x 4             

=  64

On using above property this problem  can be also solved in the following way;

 ( 2 ² ) ³ 

=  2 ⁶               

= 2 x 2 x 2 x 2 x 2 x 2             

=  64

On the basis of the above laws of Powers and Exponents, We can optimize our time to solve problems not only in algebra but in all branches of pure sciences as well as in the fields of engineering sciences.

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