This question comes from ch.7.3 titled :Adding and subtracting rational Expressions with the same denominator and least common denominator.
What is the LCD of 1/10x^4 and 4/5x
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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
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
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.