LCD with rational expressions

When we looked at how to find the lowest common denominator we learnt how to add and subtract fractions. To find the lowest common denominator (LCD), we need to find the lowest common multiple between the denominators.

For example, let's say we want to find the LCD between $\frac{1}{4}$14​ and $\frac{3}{10}$310​. We could multiply the denominators together to get a denominator of $40$40. However, this is not the lowest common denominator.

We can look at factors that the terms already have in common, then multiply the terms 

$\frac{1}{4}=\frac{1}{2\times2}$14​=12×2​ and $\frac{3}{10}=\frac{3}{2\times5}$310​=32×5​

So $2$2 is a common factor. Now we can take this common factor and multiply it by any uncommon factors from the terms to find our LCD.

$2\times2\times5=20$2×2×5=20, so $20$20 is our lowest common denominator.

 

Finding the LCD between Algebraic Fractions

We can use this same process to simplify algebraic fractions. Let's look at an example:

Find the LCD for $\frac{12}{4x}$124x​ and $\frac{6}{20x^3}$620x3​.

1. Find the LCM between the coefficients.

Our coefficients are $4$4 and $20$20. $4\times5=20$4×5=20 and $20\times1=20$20×1=20, so the LCM is $20$20.

2. Find the LCM of the variables.

Our variables are $x$x and $x^3$x3. $x\times x^2=x^3$x×x2=x3 and $x^3\times1=x^3$x3×1=x3, so the LCM is $x^3$x3.

3. Multiply the LCMs together.

The lowest common multiple is $20x^3$20x3.

 

Examples

Question 1

Question 2

Question 3