Reduce the square roots by factoring out perfect squares, if any
Make sure the denominators are rational (i.e. leave no square roots in denominators).
The order in which these are done is not important. But you can save time by examining your expression and seeing if there are advantages to one way or another.
In your expression, we could start with simplifying the square root in the denominator (because 4, which is a perfect square, is a factor in 8. But after we're done with that we'd still have a square root in the denominator. This isn't bad but it means we're not saving any time by simplifying the square root in the denominator first.
Let's look at rationalizing the denominator. To do so we will multiply the numerator and denominator by any expression that makes the denominator a perfect square. The first possibility that comes to mind is to multiply . But there is a simpler possibility: . Multiplying the numerator and denominator by this will not only rationalize the denominator but it will also leave a square root in the numerator that cannot be simplified. In other words, this one step will accomplish everything that needs to be done:
Remember, this was just the quickest way to the answer. Multiplying by works, too, but you have to simplify the numerator afterwards. And we could start by simplifying the denominator before rationalizing it. But no matter what path you take to this problem, the answer above is the only fully simplified answer with a rational denominator.
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