0

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

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

Please put parentheses around the denominators.
Are they (1/10) x^4 and (4/5) x?
Or 1/(10x^4) and 4/(5x)?

### 2 Answers by Expert Tutors

Philip P. | Effective and Affordable Math TutorEffective and Affordable Math Tutor
5.0 5.0 (428 lesson ratings) (428)
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

Parviz F. | Mathematics professor at Community CollegesMathematics professor at Community Colle...
4.8 4.8 (4 lesson ratings) (4)
-1
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.