Simplify fully (1 + √2)(4 − √2), giving your answer in the form 𝑎 + 𝑏√2, where 𝑎 and 𝑏 are integers.

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### Video Transcript

Simplify fully one plus root two times four minus root two, giving your answer in the form 𝑎 plus 𝑏 root two, where 𝑎 and 𝑏 are integers. So what they’re telling you to do basically is multiply these brackets together. And we’ve just seen an example of that using the FOIL method, but they’re asking for the answer in a very specific format. And they do intend to throw these kinds of little twists into trying to throw you off the sense sometimes. 𝑎 plus 𝑏 root two, so this just means an integer — so a whole number — plus some whole number times the square root of two. So they’re not asking you for the value of 𝑎 or 𝑏 particularly, they’re just asking you to represent your answer in that kind of layout. Okay, let’s write out the question and stop multiplying out the brackets.

So we’re gonna do one times four, which gives us four. We’re then gonna do one lot of negative root two, which is just negative root two. Then root two times four or four lots of root two, they’re both positive number; so it’s gonna be a positive answer. And then we’ve got positive root two times negative root two. So positive times negative is gonna give us a negative answer and then we’ve got root two times root two. So I just really need that for the moment, but we do know that root two times root two as we just saw is just two. So we’ve got four take away two in terms of just normal rational numbers, and four take away two is two. And then we’re starting off with negative root two. Well that really means there’s just one of them. So it’s really negative one root two, one lot of root two. And then we’re adding another four lots of root two. So if we start off a negative one on the number line and add four, we’re gonna go one, two, three, four steps up to positive three.

So here’s our answer then, two plus three root two. And this tallies with the format that we’ve been asked to give the answer in. So in this case, 𝑎 would be two and 𝑏 is the multiplier of the root two here would be equal to three. Just check the signs carefully. So we were asked for 𝑎 plus 𝑏 root two. Well we’ve got two plus three root two. So we know that 𝑎 will be two and 𝑏 will be three. As I said before, they didn’t actually ask for the value of 𝑎 and 𝑏. But in some questions, they do; so at least you now know how to answer those sorts of questions as well.