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Indices

If a number a is multiplied with itself n times (i.e., in continuous products a × a × a ... n times), where n is a positive integer, then it is called nth power of a and it is denoted as an, where n is said to be the exponent or index of a and a is called the base.

Laws of Indices

  • Description: 11544.png
     
    Example
    Calculate 22 × 23.
    Solution
    22 × 23 = 22+3 = 25 = 32
  • Description: 11538.png
     
    Example
    Calculate (32)3 × 3.
    Solution
    (32)3 × 3 = 3(2x3) × 3 = 36 × 3 = 2187 
  • Description: 11532.png
     
    Example
    Find the value of Description: 11526.png
    Solution
    Description: 11520.png 
  • Description: 11514.png
     
    Example
    Find the value of Description: 11508.png
    Solution
    Description: 11502.png
     
    Description: 11496.png 
  • Description: 11490.png
     
    Example
    Calculate (4 × 3 × 2)2.
    Solution
    (4 × 3 × 2)2 = 42 × 32 × 22 = 16 × 9 × 4 = 576 
  • Description: 11484.png
     
    Example
    Calculate Description: 11478.png
    Solution
    Description: 11471.png 
  • Description: 11464.png
     
    Example
    Calculate Description: 11458.png
    Solution
    Description: 11452.png 
  • a0 = 1, where a  0

Note: Anything raised to the power zero is equal to unity.

 

Example: 40 = 1
Some variations in the above rule:
-40 = -1 [Here, it is - (40)]
(-4)0 = 1

  • If am = an, then m = n
  • If am = bm, then a = b

Example
Find the value of x, if Description: 11405.png
Solution
Given: Description: 11399.png
Let us try and reduce the expression to the simplest possible form.
Description: 11393.png 
 
Example
Simplify: Description: 11387.png
Solution
Try to reduce the expression to a simple form by combining terms with the same base.
Description: 11381.png 

 

Example
Find the value of Description: 11373.png
Solution
Description: 11367.png
Writing the above expression in a form where the bases are same, we get,
Description: 11361.png 
 

Example
Simplify Description: 11355.png
Solution
Description: 11348.png.
Description: 11342.png 
Description: 11336.png Description: 11330.png 
 
Example
Simplify Description: 11324.png
Solution
We should try to combine terms with same base and cancel out any common factors.
Description: 11318.png 

 

Example
If Description: 11312.png then n = ?
Solution
Given: Description: 11306.png
Description: 11299.png





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