Simplifying surds (Higher)

Say a square courtyard measures five metres along each edge. Walk it corner to corner, and the diagonal comes out as the square root of fifty metres. Root fifty. As a decimal, seven point zero seven one and on forever. But root fifty is exactly five root two - five copies of root two, the diagonal of a one metre square. Same number, tidier form, and the tidy form shows what the number is made of. Turning root fifty into five root two is called simplifying a surd. That skill is this whole video.

It's a Higher tier topic. Sitting Foundation, it won't be on your paper - skip it with a clear conscience. On Higher, good news: the whole thing runs on two facts and one habit.

First, what a surd actually is.

A surd is a root that refuses to come out exactly. Root two starts one point four one four two one, and the digits never stop and never repeat. So the rule this whole chapter runs on hits full force here: decimals leak. One point four one leaks; root two doesn't. Keep the root sign, and you keep the exact number. Not every root is a surd, though. Root nine is exactly three. It comes out clean, so it isn't a surd at all.

Quick check, and you can mark it yourself. Three roots: root sixteen, root ten, root twenty-five. Which one is the surd?

Have a think. I'll wait. The answer is root ten. Root sixteen is exactly four, and root twenty-five is exactly five - both come out clean. Root ten starts three point one six and runs on forever. That one's the surd.

Now, the method.

Fact one: multiplication passes through a root sign. The root of four times three is the same as root four times root three. One root splits into two. Fact two: if one of those pieces is a square number, its root is a whole number - and a whole number doesn't need a root sign. It steps outside. Watch both facts work on root fifty. Fifty is twenty-five times two, and twenty-five is a square number. So root fifty splits into root twenty-five times root two. Root twenty-five is exactly five. The five escapes the root sign; the two stays inside, because there's nothing square about it. Five root two. That's the whole move: find a square factor, split the root, and let the square factor's root walk out. Which means the skill underneath is spotting square factors: four, nine, sixteen, twenty-five, thirty-six, forty-nine. If one of those divides the number under your root, you can split it.

Time to stress-test it.

Simplify root one hundred and eighty, giving your answer in the form a root b, where b is as small as possible.

Take your time. I'll wait right here. Here's the trap first. Four divides one hundred and eighty. Split it that way and you get root four times root forty-five - two root forty-five. Correct working. Not finished.

Look at what's still under the root. Forty-five is nine times five, and nine is square. So two root forty-five splits again: two times three root five. Six root five. That's the habit: after you simplify, check again. If the number left under the root still has a square factor, the job isn't done. Only when nothing square is left can you stop. Or save the second pass: grab the largest square factor straight away. Thirty-six divides one hundred and eighty, and thirty-six times five reaches six root five in one step. Hunt the biggest square first; keep check again as the safety net.

Next, multiplying two surds.

The split rule runs backwards too. Root a times root b equals the root of a times b. Two roots merge into one. Try root three times root twelve. Multiply under the root: three times twelve is thirty-six. So the product is root thirty-six. And root thirty-six is exactly six. Two surds multiplied, and the root sign vanishes. Multiply first, then simplify what lands - sometimes it goes all the way down to a whole number. Adding takes one line: like surds collect like terms. Two root three plus five root three is seven root three - the root three behaves exactly like the x in two x plus five x. One warning. Simplifying a surd is not cancelling a fraction. You can't just divide the number under the root by two - that changes its value. The only thing that ever leaves a root sign is a square factor.

Before the recap, a close look at how exams word this.

Exam questions usually say: give your answer in the form a root b, where b is as small as possible. That last phrase does real work. Two root forty-five is in the form a root b - but forty-five isn't as small as possible, so it doesn't answer the question. Six root five does. And when a question says exact answer, the surd stays in. Five root two is exact. Seven point zero seven is a rounded decimal - it leaks. Your turn, then - the same check the exam will run. Simplify root seventy-two into the form a root b, and make sure nothing square is left under the root.

Take your time. I'll wait right here. The answer is six root two. Seventy-two is thirty-six times two, so root seventy-two is root thirty-six times root two. Split out four instead and you reach two root eighteen - that's your check-again moment, because eighteen is still hiding a nine.

One last pass, top to bottom. A surd is a root that never comes out exact. Keep it as a root, because decimals leak. To simplify, find a square factor and split: root fifty is root twenty-five times root two, which is five root two. Then check again until nothing square is left - that's how two root forty-five becomes six root five. And to multiply, merge the roots: root a times root b is the root of a times b. Multiply, then simplify what lands.

Next in the chapter: Rationalising the denominator. It stays Higher tier, and it leans directly on the simplifying you've just learned.

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