Simplifying Complex Rational Expressions
Imagine a fraction within a fraction—sounds complicated, right? That’s exactly what a complex rational expression looks like! It could have fractions in the numerator, the denominator, or even both. But don’t worry, simplifying them isn’t as tricky as it might seem.
To make these expressions easier to work with, your goal is to combine everything in the numerator into one single fraction, and do the same for the denominator. Once you have one fraction over another, you can simplify the expression just like you would divide any two fractions: by multiplying the top fraction by the reciprocal (or the flip) of the bottom fraction.
- Combine the expressions in the numerator into a single rational expression by adding or subtracting.
- Combine the expressions in the denominator into a single rational expression by adding or subtracting.
- Rewrite as the numerator divided by the denominator.
- Rewrite as multiplication.
- Multiply.
- Simplify.
Step 1: Rewrite the Numerator
The numerator [latex]a[/latex] can be expressed as [latex]\dfrac{a}{1}[/latex] to facilitate operations.
Step 2: Combine Expressions in the Denominator
The denominator contains [latex]\dfrac{1}{b}+c[/latex]. Find a common denominator, which is [latex]b[/latex], resulting in: [latex]\dfrac{1+bc}{b}[/latex].
Step 3: Rewrite as Multiplication
Use the reciprocal of the combined denominator to convert the division into multiplication:
[latex]\dfrac{a}{1}\cdot \dfrac{b}{1+bc}[/latex]
Step 4: Simplify
The multiplication of the fractions simplifies to:
[latex]\dfrac{ab}{1+bc}[/latex]