Adding, subtracting, multiplying and dividing rational expressions
How to find the least common denominator, add and subtract rational expressions, multiply and divide them, raise them to powers and keep track of excluded values.
Adding and subtracting
It works just like numerical fractions: rewrite everything over a common denominator. The best choice is the least common denominator (LCD), the LCM of the denominators once they're factored.
Multiplying
Canceling before you multiply keeps the numbers small and saves a lot of work.
Dividing
Powers
Raise both the numerator and the denominator to the power:
\[\left(\frac{A}{B}\right)^n = \frac{A^n}{B^n}\]
Longer expressions
Follow the usual order of operations. A practical tip: factor every polynomial right at the start. You'll write the excluded values once, and the cancellations will jump out at you.
Practice
Quick check
What is the LCD of \(\frac{1}{x^2 - 9}\) and \(\frac{1}{x^2 - 3x}\)?
Factored, the denominators are \((x + 3)(x - 3)\) and \(x(x - 3)\). The LCD uses each factor once: \(x(x + 3)(x - 3)\). Multiplying the two denominators would work too, but it isn't the least.
Quick check
For \(\frac{x}{5} \div \frac{x + 6}{2}\), which value must be excluded?
Flipping the divisor puts \(x + 6\) in a denominator, so it can't be zero.
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