Rationalising the denominator (Higher)

Four divided by root three. Two students tidy it up. The first rubs out the root sign and hands in four over three. The second moves the root up top and hands in four root three over three. Both answers look neat. They are not the same number. Four over root three is about two point three one. Four over three is one point three three. Rubbing out a root sign does not simplify a fraction, it changes its value. But there is a legal way to get a root off the bottom, and it takes one move. It is called rationalising the denominator.

One flag before we start: this is a Higher tier skill. If you are sitting Foundation, you can skip this video with a completely clear conscience. If you are on Higher, good news: this is one of the quickest skills in the chapter to get solid.

First, why the bottom of a fraction gets special treatment at all. Across maths, a fraction with a root in its denominator, the bottom, does not count as fully simplified. Four over root three is correct but unfinished, and questions will ask for the finished form. Why the fuss? Dividing by a root is horrible. Root three is one point seven three two, digits never ending, so you would be dividing by a decimal that never settles. Dividing by a plain three is easy. And this chapter's warning applies here too: decimals leak. Round that root and your answer drifts. So we never turn the root into a decimal, and we never delete it. We relocate it.

The method runs on one fact about roots, one you already used in the video on simplifying surds. Quick check that it is still there.

Root five times root five. Is it option A, five. Option B, twenty five. Or option C, two root five?

Pick one. You have a moment. It is option A: five. A square root times itself gives back the number under it, because that is what a square root means. So root three times root three is three. That one fact powers this entire video.

Now the move. Look at root three over root three. Top and bottom are identical, so that fraction is exactly one. And multiplying by one changes nothing. So multiplying four over root three by root three over root three leaves its value untouched. We have multiplied by a fancy one.

That is your handle for this video: multiply by a fancy one. Never delete the root. Multiply top and bottom by it, and watch it move upstairs.

Time to run it properly, on the question from the opening. Write four over root three in the form a root three over b, where a and b are integers. The bottom is root three, so our fancy one is root three over root three. Bottom first. Root three times root three is three. The root has left the denominator. That is the whole point of the move. Now the top, and this is its own step, never an afterthought. The top gets multiplied by root three as well. Four times root three is four root three. Skip this and you have only multiplied the bottom. That is not multiplying by one any more, so the fraction you write down is a different number. Together: four root three over three. Same value as four over root three, about two point three one either way, but the denominator is now rational. The root did not die. It moved upstairs.

Next wrinkle. Sometimes the root leaves the bottom and the answer still is not finished. This one is yours first. Rationalise the denominator of six over root two, and give your answer in its simplest form. Pause and have a go. The fancy one gets you started. Multiply top and bottom by root two. Top: six times root two is six root two. Bottom: root two times root two is two. So we have six root two over two. If you stopped there, that is the trap in this question. No root on the bottom does not mean finished. Six over two cancels, so six root two over two is three root two. The root just watches while the numbers cancel. So the routine has a final step. Rationalise, then check for cancelling. Simplest form means both jobs done.

One more wrinkle and you have the full method: a number sitting in front of the root on the bottom. Rationalise the denominator of ten over two root five, and give your answer in its simplest form. The denominator is two root five. That is two, times root five. Here is the decision that matters. The two is already rational. It is not the problem. The only troublemaker is root five. So multiply top and bottom by root five alone, not by the whole two root five. Top: ten times root five is ten root five. Bottom: two root five times root five. The roots pair up into five, and five times two is ten. So we have ten root five over ten. Ten over ten cancels all the way down. The answer is just root five. Could you have multiplied by the whole two root five? It works, since two root five over two root five is also one. But the numbers balloon: twenty root five over twenty, more cancelling, more chances to slip. Just the root keeps the arithmetic small.

Before the exam corner, one full question with everything in it. This one is all yours. Rationalise the denominator of eight over root two, and simplify your answer fully. Pause the video and work it through. I will wait. Fancy one: root two over root two. Top: eight root two. Bottom: two. That is eight root two over two, and eight over two cancels to four. The answer is four root two. If you got that, you have just done the whole skill, start to finish.

Now the exam corner. What examiners say they actually see when this topic comes up. Start with the deleted root, the wrong method from our opening. It is not a rare slip. Here is AQA's examiner report from a Higher paper: A small proportion of students were clearly proficient in simplifying and rationalising surds and quickly arrived at the correct answer. The majority, however, made little or no progress, often simply removing root signs from numbers without rationalisation. The majority. Removing root signs, confidently, all the way to a tidy wrong answer. You know why it fails: deleting a root is not multiplying by one, so it changes the value of the fraction. The other thing they see comes from OCR's report on a rationalising question: Most candidates knew that they needed to rationalise the denominator, although many had forgotten how to do this and multiplication of numerator and denominator by root two, or by three minus root two, were quite common. Knowing the word rationalise is not the method. What earns the marks is the multiplication itself, top and bottom, by the root from the denominator. So write the top line in your working, four times root three, as its own step. That is the line that goes missing.

Here is the whole skill again, in four lines. One: a root on the bottom means you are not finished. Two: multiply top and bottom by that root, the fancy one, so the value never changes. Three: the root times itself becomes a whole number, and the root itself moves to the top instead of vanishing. Four: check for cancelling at the end, and when the bottom is something like two root five, only the root needs the treatment. And the chapter's thread one final time: decimals leak, exact forms do not. Rationalising keeps your answer exact and finished at the same time.

That completes our chapter on exact values and surds. Fractions kept exact, answers left in terms of pi, surds simplified, and now denominators rationalised. The full set.

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