What is the LCD of 1/10x^4 and 4/5x

Question

This question comes from ch.7.3 titled :Adding and subtracting rational Expressions with the same denominator and least common denominator.

Answer 1

1/(10x⁴) + 4/5x

 

Like any fraction, you can only add or subtract two rational expressions if they have the same denominator.  To find the LCD, you have to do a "prime factorization", factoring each denominator down to its prime numbers.

 

10x⁴ = 2*5*x⁴

5x = 5*x

 

The LCD includes each unique prime factor.  If a factor appears in more than one denominator, then choose the factor with the highest power.  So for the above case, the unique prime factors are 2, 5, and x⁴:

 

LCD = 2*5*x⁴ = 10x⁴

 

To add then, we must change the denominator of the second expression to 10x⁴ by multiplying it by the prime factors of the LCD it is missing, which is 2*x³:

 

1/(10x⁴) + (4/5x)(2x³/2x³) = 1/(10x⁴) + (8x³)/(10x⁴) = (1 + 8x³)/10x⁴

 

 

Answer 2

 LCM of 2 or more numbers with the highest exponents= product of the common factors with highest   exponents  ( 1)

 

  You know that by Fundamental laws of arithmetic every composite number is uniquely equal to the product of its prime factors

 

   Take 2 numbers 36 , 24 ,using statement (1)

      36 = 2² . 3²

 

       24 = 2³ . 3

   

       LCM ( 24, 36) = 2³ . 3² = 8 . ( 9) = 72      ( 2)

 

         If we take   A = a² b²  , and    B = a³. b

 

        LCM ( A , B ) = a³ b² , where 36, 24 is a case where a =2 and b=3

 

     Relation (1) implies that lowest number that divides both 36, 24 into it is 72

 

         then

   GCF - greatest Common factor is the greatest factor that both numbers can be divided by it.

             it is the product of the common factor with the lowest exponent.

 

             GCF ( 24, 36) = 4 . 3 = 12

 

              So  :  LCM ( A, B ) = A. B / GCF ( A, B) 

 

               LCM ( 24, 36) = (24 . 36 ) / 12 = 72.