# CLASS-6EXPONENTS

EXPONENTS

If ‘A’ is any number and m is the natural number and if we wish to multiply unlimited or m times of A each other, then we can write or express like that-

( A x A x A x A x …………… m times taken ) =  A, where ‘m’ is the exponents and ‘A’ is the base.

There are some rules of ‘EXPONENTS’ are described below which we have to follow every time in application-

1) If ‘A’  is any number and let x & y are the natural numbers, then-

Aˣ   x   Aʸ  =   Aˣʸ

2) If ‘A’ is any number, m & n are the natural number, where m > n then-

A

---------  =   A ⁽ ˉⁿ

Aⁿ

3) If ‘A’  is any number, m & n are the natural number, then

( A) ⁿ =   A

4)   If  ‘A’ & ‘B’ are any number and m is a natural number, then

A  x   B  =  (AB)

5)   If  ‘A’ & ‘B’ are any number and m & n are natural numbers, then

A  x   Bⁿ =  Aᵐ Bⁿ

6)   If ‘A’ is any number and m is a natural number, then

(-A) =  A  when ‘m’ is even.

And     (-A) = - A when ‘m’ is odd.

7)   If ‘A’ is any number, then

A =  1

A¹ =  A

Example.1)  Please write the value of 7

Ans.)  7⁴ =  7 x 7 x 7 x 7  =  2401

Step.1) First observed in given numbers how much power or natural number is given.

Step.2) Multiply normally the given number 7 as much as time as per given Power or natural numbers (i.e. 4) is provided. Here number 7 is to be multiplied by 4 times and the result is 2401.

Step.3) Value comes with negative (-) sign or value because as per rules-

A = A or (+) A when ‘m’ is even. So, 2401

or (+) 2401 is the actual/final result.  (Ans.)

Example.2) Please write the value of  (-2)

Ans.)  (-2) = (-2) x (-2) x (-2) x (-2) x (-2) x (-2) x (-2) = -128

Step.1) First observed in given numbers how much power or natural number is given.

Step.2) Multiply normally the given number 2 as much as time as per given Power or natural numbers (i.e. 7) is provided. Here number 2 is to be multiplied by 7 times and the result is 128.

Step.3) Value comes with negative (-) sign or value because -

as pr rules (-A) = - A when ‘m’ is odd. 7 is odd number

So, -128 is the actual/final result.      (Ans.)

Example.3)  Please write the value of  (-3)

Ans.)  (-3)⁸ = (-3) x (-3) x (-3) x (-3) x (-3) x (-3) x (-3) x (-3) = 6561

Step.1) First observe in given numbers how m power or natural number is given.

Step.2) Multiply normally the given number 3 as much as time as per given Power or natural numbers (i.e. 8) is provided. Here number 3 is to be multiplied by 8 times and the result is 6561.

Step.3) Value comes with negative (-) sign or value because -

as pr rules (-A) = A  when ‘m’ is even.

So , 6561 is the actual/final result. (Ans.)

4) Write 625 as a power of 5.

Step.1)  First find the prime factorization of the given number.

Step.2)  After completing prime factorization of the given number observe how many base has come with their exponents. Clearly shown that only 5 has come 4 times .

Step.3) Clearly shown that 5 has come 4 times, so 5 is the base and 4 its exponents.

So, it can be written as 5.

625 =  5 x 5 x 5 x 5 = 5.    (Ans.)

5) Write 38880 in exponential form.

Step.1)  First find the prime factorization of the given numbers.

Step.2)  After completing prime Factorization of the given number observe how many base and their exponents we can find.

Step.3) Clearly shown that 2 has come 5 times, 5 has come 1 time and 3 has come 5 times. so it can be written as 5.

Step.4) 2 has come 5 times, so 2 is base 5 its exponent. It can be written 2. 5 has come 1 time, so 5 is base and 1 its exponent. It can be written 5¹. 3 has come 5 times, so 3 is base and 5 its exponent. It can be written 3.

Step.5.) Exponential form can be written as multiplication of base with their exponents like

2 x 5¹ x 3

38880 = 2 x 2 x 2 x 2 x 2 x 5 x 3 x 3 x 3 x 3 x 3

= (2 x 2 x 2 x 2 x 2) x 5 x (3 x 3 x 3 x 3 x 3)

= 2 x 5¹ x 3  (Ans.)

6)  Write the product of number 3 and 3 as single power.

Ans.)   3 and 3

=  3 x  3 =  3 ⁺⁵ =  3

Step.1)  Observe the given two-digit, where the base are the same but exponents are different.

If ‘A’  is any number and let x & y are the natural numbers,

then-  Aˣ x Aʸ  =  A⁽ˣʸ

3 =  3 x 3 x 3 x 3 x 3 x 3 x 3 x 3 6561    (Ans.)

7)  Write the product of number (-8) and (-8) as single power.

Ans.)   (-8) and (-8)

=  (-8) x (-8) =  (-8) ⁵⁺⁷ =  (-8)¹²

Step.1)  Observe the given two-digit, where the base are the same but exponents are different.

Step.2) We follow the rules- If ‘A’ is any number and let x & y are the natural numbers,

then-  Aˣ  x  Aʸ  =  A⁽ˣʸ

= (-8) x (-8) x (-8) x (-8) x (-8) x (-8) x (-8) x (-8) x (-8) x  (-8) x (-8) x (-8)

=  (- 68719476736)  (Ans.)

8)  Evaluate   (3

(3)²  =  (3)⁴˟² =  3

Step.1) We can observe that, there is one base two exponents

Step.2)   We will follow the rules –

If ‘A’ is any number, m & n are the natural number, then-

( A) ⁿ =   A

3 =   3 x 3 x 3 x 3 x 3 x 3 x 3 x 3  = 6561   (Ans.)

9)  Multiply 5 and  2

Ans.)   5 and  2 =  5  x  2  =  10

Step.1) We can observe that, in given numbers base are different but exponents are equal.

Step.2) we follow the rules as discussed below -

If ‘A’ & ‘B’ are any number and m is a natural number, then

A  x  B  =  (AB)

=  10  =  10 x 10 x 10 x 10

10000           (Ans.)

10) Explain (-5) = ?

Ans.)  (-5)

=  (-5) x (-5) x (-5) x (-5) x (-5)

= - 3125     (Ans.)

Step.1) Here we can observe that there are single base and the exponents are shown 5 times, so we have to multiply the given base by 5 times.

Step.2) Value comes with a negative (-) sign or value because -

as pr rules (-A) = - A when ‘m’ is odd. 5 is odd number

(-5) x (-5) x (-5) x (-5) x (-5) - 3125

So , -3125 is the actual/final result.     (Ans.)

11)  Evaluate  9 / 9

Ans.)    9 / 9

9

= ---------   =    9⁷⁻⁴   =   9

9

Step.1) Observe in a given fraction base are the same but exponents are different. so

Step.2) We follow the rules -

If ‘A’ is any number, m & n are the natural number, where m > n then-

A

----------  =  A ⁽ᵐˉⁿ

Aⁿ

=  9ᶟ  =  9 x 9 x 9   =    729     (Ans.)