Identify which arrow represents the position of root 17 on the number line.
So we have 𝑎 right by four and 𝑏 by five. So to work out which arrow represents the position of root 17, what we can do is in fact square both four and five. So if we square four, we’re gonna get 16 because four multiplied by four is 16. And if we square five, we’re gonna get 25. Well, therefore, as 17 is much closer to 16 than it is to 25, we can say that the arrow 𝑎 must represent the position of root 17 on the number line.
However, even though we found the answer, cause it does look sensible because as we said 17 is much closer to 16 than it is to 25, what we can do is double-check this by squaring 4.5. And the reason we’re going to do this is because if we square 4.5 and find out what the result is, if 17 is less than this number, then 𝑎 is the correct position. However, if 17 is bigger than this number, then 𝑏 will be the correct position.
So what we can do to multiply 4.5 by itself is in fact multiply 45 by 45. Then we can put the decimal places back in afterwards. So we have five multiplied by five, which is 25. So we carry the two. And then five multiplied by four is 20. So we get 225. So then next we’re gonna multiply by the four or the 40 here. And as it’s a ten, what we’re gonna do is put a zero into the units column. Then we’ve got four multiplied by five, which is 20. So we put a zero in the tens and carry two. And then four multiplied by four is 16, add the two we’ve got, which is 18, so we’ve got 1800. Then we add these together, which gives us 2025.
Well, now if we look at the original multiplication, it was 4.5 squared, so 4.5 multiplied by 4.5. And we can see here there are two numbers after the decimal point. So therefore, our answer must have the same. So 4.5 squared is equal to 20.25. And as 17 is less than 20.25, we can confirm that yes, 𝑎 does represent the position of root 17 on our number line.