Calculate the LCD as the product of the prime factors of the denominators, each raised to the greatest power that occurs in the factored denominators.
Let's take a look at a few examples showing how this works.
Numerical Fractions
1^(st) Denominator Prime Factor(s)
2^(nd) Denominator Prime Factor(s)
LCD
1/2 and 1/6
2= 2
6= 2* 3
2* 3=6
2/15 and 5/6
15= 3* 5
6= 2* 3
3* 5* 2=30
3/8 and 7/10
8= 2* 2* 2
10= 2* 5
2* 2* 2* 5=40
Finding the LCD of Two Rational Expressions
Recall that a rational expression is a fraction whose numerator and denominator are polynomials.
Rational Expression:polynomial/polynomial
To find the LCD of two rational expressions, we complete the same steps as when finding the LCD of two numerical fractions.
Write each denominator as a product of its prime factors.
Calculate the LCD as the product of the prime factors of the denominators, each raised to the greatest power that occurs in the factored denominators.
Note that, in this case, the denominators may contain variables. Therefore, their prime factors might not only be prime numbers but also polynomials that cannot be further factored. Here are a few examples.
Rational Expressions
1^(st) Denominator Prime Factor(s)
2^(nd) Denominator Prime Factor(s)
LCD
1/4 and 3/2x
4= 2* 2
2x= 2* x
2* 2* x=4x
x-7/2x^2 and x+1/6x
2x^2= 2* x* x
6x= 2* 3* x
2* x* x* 3=6x^2
x^2-3/x^3+x and 5/x^2+x
x^3+x= x*( x^2+1)
x^2+x= x*( x+1)
x*( x^2+1)*( x+1)=x^4+x^3+x^2+x
Conclusion
Considering all of the above information, we can see that the steps we take to find the LCD of two numerical fractions and the steps we take to find the LCD of two rational expressions are identical. The difference is that the denominators of rational expressions may contain variables whereas the denominators of numerical fractions never contain variables.
Expression
Numerical Fraction?
Rational Expression?
3/4
Yes âś“
Yes âś“
x^2+1/5
No *
Yes âś“
2/x-1
No *
Yes âś“
In fact, any real number can be viewed as a monomial which is a type of polynomial. Therefore, any numerical fraction is also a rational expression. This is why it makes sense that finding the LCD of two numerical fractions is not that different from finding the LCD of two rational expressions.
