## anonymous one year ago Can a variable be a least common denominator if it is the only common denominator?

1. mathstudent55

$$\dfrac{5y}{2x^2} + \dfrac{3z}{3x}$$ The only common factor of the denominators is x. x is not the LCD. $$\dfrac{y}{x} - \dfrac{z}{x}$$ x is the the only common factor of the two denominators, and x is the LCD.

2. anonymous

then what would the LCD be of the first expression?

3. anonymous

@mathstudent55

4. mathstudent55

To find the LCD, use common and not common factors with the largest exponent. I'll explain what that means.

5. mathstudent55

What is the LCD of 180 and 108? First, we find the prime factors of each number. $$180 = 2^2 \times 3^2 \times 5$$ $$108 = 2^2 \times 3^3$$ As you can see, both 180 and 108 have 2 and 3 as factors. 180 also has 5 as a factor, but 108 does not have 5 as a factor. To find the LCD of 180 and 108, we follow the rule. We need common and not common factors with the largest exponent. We have common factor 2 appearing as 2^2, and 2^2. Pick 2^2 (2 is the only exponent of factor 2, so pick it). Then we have common factor 3^2 and 3^3. Pick 3^3 (the larger of the two exponents) Finally, there is not common exponent 5, which only 180 has. Pick it also. LCD = 2^2 * 3^3 * 5 = 4 * 27 * 5 = 540 The LCD of 180 and 108 is 540.

6. mathstudent55

Now let's look at my example with variables above. What is the LCD of 2x^2 and 3x? 2x^2 = 2 * x^2 3x = 3 * x LCD = 2 * 3 * x^2 = 6x^2

7. anonymous

Thanks so much!