Question

Can a variable be a least common denominator if it is the only common denominator?

Answer 1

\(\dfrac{5y}{2x^2} + \dfrac{3z}{3x}\)
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.

Answer 2

then what would the LCD be of the first expression?

Answer 3

@mathstudent55

Answer 4

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

Answer 5

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.

Answer 6

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

Answer 7

Thanks so much!