Why do we have to simplify the fraction when simplifying it still contain a square root?

victorioussheep

I don't get why we have to simplify it when there would still be a square root in the simplified fraction?

quacker88

If I'm not mistaken, you're asking why we usually prefer writing square roots as \( \frac{\sqrt3}{3} \) instead of \(\frac{1}{\sqrt3}\), even though they are the same thing, right?

Here's the thing, both of them have their uses. For example, if I told you to multiply that number by \(\sqrt3\), it would probably be easier to do \(\frac{1}{\sqrt3}\cdot\sqrt3\), than \( \frac{\sqrt3}{3} \cdot\sqrt3 \) even though both would get you the same answer.

But think about it this way-- what does a fraction LITERALLY mean? It's divison, right? \(\frac{6}{2}\) says, "I'm going to divide 6 into 2 parts. Each part has 3, so \(\frac{6}{2}=3\)."

But how would you do \(\frac{1}{\sqrt3}\)? How can you divide \(1\) into \(\sqrt3\) parts? \(\sqrt3\) is irrational, it has infinitely many digits after the decimal point! That's not very convenient, is it?
On the other hand, \( \frac{\sqrt3}{3} \) says "Divide \(\sqrt3\) into \(3\) parts." Even thought there is still an irrational number, it makes a lot more sense to divide it into \(3\) parts, since that's a whole number we can deal with.

Hope this helps!

victorioussheep

@quacker88 Thank you for your response! It helps a lot!