EXPONENTS

 

    Exponents is defined as the method of expressing large numbers in terms of powers . That means, exponents refers to how many time a number multiplied by itself . For example 5 is multiplied by itself by 6 times , i.e. 5 × 5 × 5 × 5 × 5 × 5 . This can be written as 5⁶ here, 6 is the exponent and 5 is the base . 

Exponent symbol  

     The symbol used for representing the exponent is ^. This symbol (^) is called a carrot. For example, 6 raised to 3 can be written as 6^3 or 216. Thus, 6^3 = 6 × 6 × 6 = 216.

some example .,   

  1.     4×4×4 = 4³  .... Here 4 is the base and 3 is exponent . 
  2.     3×3×3×3×3 = 3⁵   ...... Here 3 is the base and 5 is exponent. 
Exponent Laws 

1. Multiplication law :  If base are same then add thier exponents and calculate ex.,       3² × 3⁴ here base is same and their exponents are different so, first add the exponents                i.e  32+4    =  3⁶   =   729                                                                                                                        
     
 If base are not same then find their exponent value and multiply ex.,                              2³ ×     4² = 8 × 16 =128

 2. Division law :   Bases – dividing the like ones; subtract the exponent of the denominator from the exponent of the numerator Exponent and keep the base the same.

Let ‘a’ be any integer or a decimal number and ‘m’, ‘n’ are positive integers, that represent the powers to the bases such that the above laws can be written as:
  • aᵐ.aⁿ = am+n
  • (aᵐ)ⁿ = amn
  • (ab)ⁿ = aⁿbⁿ
  • (a/b)ⁿ = aⁿ/bⁿ
  • aᵐ/aⁿ =  am-n
  • aᵐ/aⁿ = 1/ an-m

Exponent and Powers

As defined above, the exponent defines the number of times a number is multiplied by itself. The power is an expression that shows repeated multiplication of the same number or factor. For example , in the expression  5⁶ , 6 is the exponent and  5⁶ is called the 5 power of 6 . That means 5 is multiplied by itself 6 times . 

Exponent Rule 

Zero Rule :    Zero exponent of a variable is one , i.e. power is zero of any number is equal to 1  x⁰  = 1      ex, 5⁰ = 1.

One Rule :     power is 1 of any number is equal to itself number x¹  = x        ex., 5¹ = 5

Negative Rule:     Negative exponent of a variable can be written as follows.
                          
                             x-p = 1/xp

Product Rules :   
(i) Product of exponents on same base results in sum of exponent on that base.
           
            xp.xq = xp+q
                           
  (ii) Product of same exponent on different bases is equal to that exponent  raised to whole product of the bases.
        
              xp.yp = (xy)p

Quotient Rules: 
   (i) Division of two exponents having same base results in difference of                                    exponents (numerator exponent – denominator exponent) raised to that  base.
           
           xp/xq = xp-q
                             
  (ii) Division of same exponent on different bases is written as the              exponent raised to division of the bases.
             
          xp/yp = (x/y)p

Power Rules
  Power rules are given below.
  • (xp)q = xpq
  • x1/p = p√x
  • xp/q = q√xp







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