Indices [Concept, Problems on Indices with Solutions]

Indices is one of the important math topics. It deals with numbers with exponential powers. 

This post discusses the following

• What do you mean by Indices ?

• What is a base in indices?

• Laws of Indices / rules of Indices

• Indices problems with solutions

What do you mean by Indices ?

The meaning of Index ( plural form is Indices) which means exponential power like for example

5³,  7³  etc.

This topic will deal with the formula for indices which help in calculations of those numbers which have exponents. There are laws which needs to be followed. These laws are called Laws of Indices.

Lets understand these laws of Indices and solve few problems in this blog.

What is a base in indices?

In the above explanation of Indices we have taken two examples 5³,  7³ .

Here base for 5³ is 5 , and 

base for 7³ is 7

So the number which have a exponential power is called a base.

Laws of Indices and Indices problems examples

(Consider d,h, n denote numbers )

 1 )   If there are two numbers with exponential values and these two powers are equal then both  

the bases of these two numbers are equal

Example :   If  d³ = 5³   then as powers are equal ,

  d = 5

Here we need to note, If the powers are odd numbers then the bases will have two values , positive and negative

for example -  If  d⁴ = 5⁴ ,

In this case d can be 5 or (-5) as   5⁴ and (-5)⁴  give same values.

2)  If there are two numbers with exponential values and the bases of these two numbers are equal then both  the powers of two numbers are equal

Example :   If   5³ =  5ⁿ    then as bases are equal ,

  n = 3

3)  For any number, except number '0' (Zero) , 

if its power is Zero, then the value is '1'

Example :

d⁰  = 1

40⁰  = 1

5008⁰  = 1

What is the value 0⁰ ?

It is undefined meaning , no value can be assigned.

4)  For any number ' d ' ,  if its power is Zero, then the value is 'd' 
d¹ = d
Note this is true for negative values and Also for 0 ( Zero) 

For example :  

•  25¹  = 25 , 
•  (-5)¹ = -5 ,  
•  0¹ =  0

5)  Lets d be any number , then 

 

d^m x  dⁿ  = d^m+n

Example:  

•  d⁵ x  d³   = d⁵⁺³   

                    = d⁸

•  2⁵ x  2³  = 2⁵⁺³   

                    = 2⁸   

6)   d^m / dⁿ   = d^m-n
 

Example: 

  
10⁵ / 10²   = 10^5-2      = 10³               

7)   1/ d^-n    = dⁿ

Example

1/ 15^-3   = 15³

 

Similarly,

1/ dⁿ    = d^-n

Example

1/ 12³   = 12^-3

8)  ( d^m )ⁿ = d^mn

Here,  " mn " means ( m x n )

Example

( 16² )³ = 16^2x3    

9 )  m√d = d^1/m

Example:

3√27 = 27^1/3

10)  (dh)^m  = d^m x h^m

Example

(7 x 3)³  = 7³ x 3³

11)  ( d / h)ⁿ = dⁿ / hⁿ

Example :

 ( 6 / 8)⁴ = 6⁴ /8⁴

         Problems on Indices

Question 1:  Simplify this Problem  

            (36⁰ x 36² )  - 125 ?

Solution:

(36⁰ x 36² )  - 125 

= (1 x  1296 ) - 125

= 1296 - 125

= 1171

===========================

Question 2:  Simplify this Problem  

 (2 x 6⁰ x 16² )  - ( 1 / 5^-3  )

Solution:  

(2 x 6⁰ x 16² )  - ( 1 / 5^-3  )

= (2 x 1 x 256 )  - (  5³  )

= 512 - 25

=  487

========================

Question 3:  

Simplify this Problem  (27 / 125) ^-3

Solution :

(27 / 125) ^-3

⁼  (3³ / 5³) ^-3

As 1/ d^-n    = dⁿ

= (5³ / 3³ ) ³

=  5⁹ /3 ⁹

========================

Question 4:  

Simplify this Problem  (16 / 625 )^3/2

Solution:

(16 / 625 )^3/2  

=  (2⁴ / 5⁴ )^3/2

= (2⁴ )^3/2 / ( 5⁴ )^3/2

=  (2 )^12/2 / ( 5 )^12/2

=  (2)⁶ / (5)⁶

⁼  2⁶ / 5⁶

========================

Question 5: 

 Simplify this Problem  

(81 / 625 )^3/2   x (125 / 625) ^-1

^Solution: 

(81 / 625 )^3/2   x (125 / 625) ^-1

=   ( 3⁴ / 5⁴ ) ^{3 / 2}   x   ( 625 / 125 )  

=  3⁶ / 5⁶    x  ( 5⁴ / 5³ )

=   3⁶ / 5⁶    x  ( 5^4-3 )

 =  3⁶ / 5⁶    x  (5 )

 =  3⁶ / 5^6-1

=   3⁶ / 5⁵

========================

^{Question 6 :}  

Simplify this Problem

 (81 / 2401 )^3/4   ÷ (125 / 625)^1/2  x 5^1/2

 

Solution:

  

(81 / 2401 )^3/4   ÷  (125 / 625)^1/2  x  5^1/2

 

=  ( 3⁴   / 7⁴ )^3/4   ÷  ( 5³ / 5⁴ ) ^1/2   x  5^1/2

                       

 =  ( 3³  / 7³)  ÷  5 ^{( 3/2 ) – (2)}  x 5^1/2

 

=   ( 3³  / 7³)  ÷  5^1/2  x  5^1/2

 

=   3³ / ( 7³ x 5^1/2 )  x  5^1/2

=  3³  / 7³

========================

^{Question 7 :} 

 

Simplify this Problem 

 √(25 / 9 )^d+5  =   125 / 27  

 and find the value of  'd'

Solution: 

√(25 / 9 )^d+5  =   125 / 27

 

= (25 / 9 )^d+5/2  =   5³ / 3³

 

=  (5² / 3² )^d+5/2  =   5³ / 3³

 

=  (5 / 3 )^d+5  =  ( 5/3)³

 

Going by the rule 1 mentioned in the beginning of the blog,

We can write

d + 5 = 3

d = -2

========================

^{Question 8 :}  

Simplify this Problem  

 

576^5d-2d-6  = 24. 

Find the value of ' d ' ?

Solution:

• 576^5d-2d-6  = 24

 

•  576^3d-6  = 24

 

• 24^2(3d-6) = 24

 

So going the law 1 , we can write

•      2(3d-6) = 24

•        3d – 6 = 12

•       3d = 18
•       d = 6

======================== 

Question 9 -  

Find the value of  ' d ' in the following equation 

512^5d-30  = 2^15d

^Solution:

• 512^5d-30  = 2^15d

• 2^9(5d-30) = 2^15d

• 45d – 270 = 15d

• 30d = 270

• d=270 /30 = 9

========================

Question 10 -

^{Simplify the problem}

(1296d⁴)^7/4  /( 729d²)^1/2

^Solution:

(1296d⁴)^7/4  /( 729d²)^1/2

⁼ (6⁴ d⁴)^7/4  / ( 3⁶ d²)^1/2

⁼ (6d)⁷/ ( 3³ d)

= 6⁷ d⁶ / 27