Binomial Expansion

Using the first three terms of the expansion of (1−2x)12 and the substition x=0.01, find an approximate value for 0.9812.

Give your answer to four decimal places.

Can someone show me how to work this out please

I assume that you mean (1 - 2x)^12 and 0.98^12 - I'm not one of the people here who insists on using LaTeX for everything, but you need some sort of formatting to make the questions make sense!

So, first things first - do you know how to do the binomial expansion of (1 - 2x)^12 ?

Yeah sorry, I just copied and pasted the question directly from the website but just assumed that it was the same as what I saw, oops.

Yes I do know the binomial expansion of (1-2x)^12, but I haven't had to substitute the numbers in before so am not sure how to do this.

You need to use your expansion of (1 - 2x)^12 to estimate 0.98^12. If we ignore the power of 12 (which is the same in both cases), this means that to use your expansion, you need 1 - 2x to be the same as 0.98. This leads to x = 0.01. In this question, they have been quite nice to you and have told you to use this value, but in another question you might be expected to do this yourself.

If you put x = 0.01 into the entire expansion of (1 - 2x)^12, you will get the exact value of 0.98^12. But not only is this a lot of work, most of it is a waste of time. 0.01 is a small number, so when you raise it to higher and higher powers, you will get smaller and smaller answers. Therefore, you can stop after a while, as the extra bits you are adding on don't make a lot of difference.

That is the idea here. If you only use the first three terms, that is, the constant term and the terms in x and x^2, you will get a value which isn't exactly 0.98^12, but it is pretty close. That's what they want you to do.

One thing that makes questions like this a bit odd in the modern age is that it never seems to be explained in text books why you would bother doing this. In the days before calculators, this was a good way to get approximations for calculations that would be tricky to do otherwise. I mean, how would you work out 0.98^12 without a calculator? Multiply it by itself 12 times? There are shortcuts, but it is still a lot of awkward number work. Using the start of a binomial expansion is much easier, and only requires multiplying reasonably small numbers and then adding them together. Of course, these days, anyone can just tyoe 0.98^12 into a calculator, which makes these questions a bit antiquated - they seem to be on question papers becasue the can be, rather than because anyone would actually do the calculations that way. A bit of historical background might motivate students to see why they are bothering.

One thing that makes questions like this a bit odd in the modern age is that it never seems to be explained in text books why you would bother doing this. In the days before calculators, this was a good way to get approximations for calculations that would be tricky to do otherwise. I mean, how would you work out 0.98^12 without a calculator? Multiply it by itself 12 times? There are shortcuts, but it is still a lot of awkward number work. Using the start of a binomial expansion is much easier, and only requires multiplying reasonably small numbers and then adding them together. Of course, these days, anyone can just tyoe 0.98^12 into a calculator, which makes these questions a bit antiquated - they seem to be on question papers becasue the can be, rather than because anyone would actually do the calculations that way. A bit of historical background might motivate students to see why they are bothering.