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Question 432481: I am needing help in my Algebra 2 class. Right now we are working on adding and subtracting rational expressions. I do not know how to find out the LCD of the problems. If you could help me, I would greatly appreciate it! (:
~Andrea~
Found 2 solutions by ewatrrr, J2R2R:
Answer by ewatrrr(24785)   (Show Source):
You can put this solution on YOUR website!
Hi
Re: LCD
My recommendation would be:
In order to quickly proceed with the simplification
1) Note the basic facts as to the factors in the denominators
For ex: 3/X + 4/Y = (3Y + 4X)/XY
2) |X and Y can be 'anything' for example:
Always important to proceed using basic principles:
3/4 + 4/12 does equal (3*12 + 4*4)/48 = 52/48 = 13/12
Yes, had we used 12 to begin with, rather than the product, 48:
(9 + 4)/12 = 13/12 would have been 'simpler' which is the point of the LCD
However, there is no excuse really, not to push one's pencil using the basics.
Many times the product of the denominators is just fine to use, my experience.
With experience, the LCD will "pop into one's head" more easily.
Answer by J2R2R(94)  (Show Source):
You can put this solution on YOUR website! The lowest common denominator is the smallest number which all the numbers will go into. I will touch on the highest common factors too.
Some of these numbers share a common element and only need to be used once so this is where prime factors come in.
By breaking everything down into prime factors they are at their lowest form and cannot be broken up any more so there is no chance of missing them.
Example
2 x 2 x 2 x 3 x 3 x 3 x 5 x 7 x 11
2 x 2 x 3 x 3 x 7 x 11 x 13
2 x 3 x 3 x 7 x 11 x 13
For ease of use let us write these using powers and where there is a prime factor in some but missing in others we include that factor to the power 0 since anything to the power 0 is 1. I am putting all powers in for illustrative purposes too.
We have
2^3 x 3^3 x 5^1 x 7^1 x 11^1 x 13^0
2^2 x 3^2 x 5^0 x 7^1 x 11^1 x 13^1
2^1 x 3^2 x 5^0 x 7^1 x 11^1 x 13^1
The highest common factor is found by taking the lowest power of each prime factor. Let us take the powers of 2. We have 2^1, 2^2 and 2^3.
2^1 will go into 2^2 and 2^3 but neither of these will go into 2^1, so 2^1 contributes towards the highest common factor.
The lowest common multiple (or denominator when used for fractions) is found by taking the highest power of each prime factor. Let us take the powers of 2 again. We have 2^1, 2^2 and 2^3.
2^1 and 2^2 will go into 2^3 but 2^3 wont go into either 2^1 or 2^2, so 2^3 contributes towards the lowest common multiple.
Doing it for all factors we have:
Highest Common Factor (lowest powers): 2^1; 3^2; 5^0; 7^1; 11^1; 13^0.
Lowest Common Multiple (highest powers): 2^3; 3^3; 5^1; 7^1; 11^1; 13^1
Highest Common Factor = 2 x 9 x 1 x 7 x 11 x 1
Lowest Common Multiple = 8 x 27 x 5 x 7 x 11 x 13
I hope this illustrates what the lowest common multiple or denominator is.
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