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Whenever you're trying to rationalize a denominator or numerator, you're generally going to take the same route. You multiply by the conjugate! This is actually a consequence of the irrational conjugate root theorem for polynomials (see link).
When talking about a conjugate, you're looking at flipping the sign of the second term. For example, the conjugate to `a+b` will be `a-b`. This is important because of the following relation:
`(a+b)(a-b) = a^2 - b^2`
This is the best way to combine two numbers so that you obtain their square through pure multiplication. This is important because the only wy you can modify a fraction expression is multiplication of the top and bottom by the same number/expression!
So how do we apply this to this particular problem? Let's look at it:
`(sqrt(5x) - 6)/(sqrt(5x) + 3)`
We are trying to rationalize the denominator, meaning we would like to find a way to get rid of the square root on the bottom. This is where conjugation comes in. We will multiply top and bottom by the conjugate of the denominator:
`(sqrt(5x) -6)/(sqrt(5x) + 3) * (sqrt(5x) - 3)/(sqrt(5x)-3)`
Notice we can do this because we're effectively multiplying by 1. Let's continue. The denominator will be the difference of the squares of `sqrt(5x)` and 3. The numerator will be a bit more complicated:
`((sqrt(5x)-6)(sqrt(5x)-3))/(((sqrt(5x))^2 - 3^2))`
Simplifying with FOIL:
` ``(5x -6sqrt(5x) - 3sqrt(5x) + 18)/(5x - 9)`
Simplifying the numerator:
`(5x - 9sqrt(5x) + 18)/(5x-9)`
There is your answer. The method is generally the same for each type of problem that asks you to "rationalize" the denominator or make it real (in the case it's imaginary).
The main takeaway point here: to rationalize the denominator or numerator, you need to multiply by the conjugate.
I hope that helps!
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