Squares and Square Roots
These square roots are nice and everything, but it would be nicer if we could turn them into fractions on top of it. It's like we can read your mind, right? Spooky.
To divide radicals, we divide the radicands.
Here's why this makes sense: if we multiply by itself, we will find . As a result, is a square root of . You may have already known that. It was probably on your "division of radicals" wallpaper when you were growing up.
Same deal with variables. If x and y are any non-negative numbers (so that we can take their square roots), then
Divide, simplifying if possible:
The rule for division with radicals can also be used in reverse. Check your rearview mirror before backing up first. You don't want to run over any variables. Or do you?
The rule can be used to break up an expression with a fractional radicand into a quotient of radicals with nicer radicands. If x and y are non-negative integers, then
These are both numbers from our list, so we can tell right away that this won't be too painful. We need to be able to deal with each of these numbers separately, so we rewrite this expression as a quotient of radicals and simplify from there.
Sometimes it's more helpful to rewrite the square root of a fraction as a fraction of square roots, as in the previous example. Sometimes, however, it's more efficient to do the division in the radicand. We wish we could tell you it was more cut-and-dry than that, but you'll need to use your own judgment. Wear a black robe and powdered wig if you think it'll help.
Turning this into a quotient of radicals yields .
Since 28 = 4 × 7,
If we had only done the division in the radicand first, however, we would have gotten
Well, that was much easier. We're on board with this method; how about you?
Which method do we use when? In general, if a radicand is a fraction that can be simplified to a perfect square or some other nice, even number (for example, an integer times a perfect square), simplify the radicand first. If the radicand is a fraction that doesn't simplify to anything we like, it's probably time to split up the expression into a quotient of square roots. Either way, simplifying correctly will give us the correct answer. If we wind up going the long way around every once in a while, so be it. At least we can admire the scenery along the way.