Binomial expansion with surds

a) Expand (2+√3)^5

b) Hence write (2-√3)^5 in the form a + b√3


For a) the answer is 362+209√3

For b) I guessed that it was 362-209√3 and I was right. The textbook solution just says this

(2-√3)^5 = (2+ (-√3))^5 = 362-209√3


If there was no simplification then it would make sense to me e.g.

(2+x)^5 = x^5 + 10 x^4 + 40 x^3 + 80 x^2 + 80 x + 32

therefore

(2-x)^5 = (-x)^5 + 10 (-x)^4 + 40 (-x)^3 + 80 (-x)^2 + 80 (-x) + 32

But the fact that this expansion has been simplified first confuses me in the fact that you can just replace the positive with the negative.

Like you can't do this replacement with a different surd e.g. this is clearly not right

(2-√5)^5 = 362-209√5

So does this only work if the surd changes sign? And is it something "obvious" that doesn't need to be proved because I don't immediately see why it's true.

For even powers of sqrt(5), its no longer a surd and only odd powers contribute to the bsqrt(5) part. So thats why you can simply flip the sign as (-1)^even is positive and (-1)^odd is negative.

Just work through a simpler example like
(1+/-sqrt(2))^n
for n=1,2,3? You should see the pattern.

Thanks, I understood this by doing the expansion and I could see why it works. But I think my question is more about if it's okay to quote the result without working?

Like could this be written in an A Level exam and get full marks?

(2-√3)^5 = (2+ (-√3))^5 = 362-209√3

Is it "obvious" enough?

a) Expand (2+√3)^5

b) Hence write (2-√3)^5 in the form a + b√3


For a) the answer is 362+209√3

For b) I guessed that it was 362-209√3 and I was right. The textbook solution just says this

(2-√3)^5 = (2+ (-√3))^5 = 362-209√3


If there was no simplification then it would make sense to me e.g.

(2+x)^5 = x^5 + 10 x^4 + 40 x^3 + 80 x^2 + 80 x + 32

therefore

(2-x)^5 = (-x)^5 + 10 (-x)^4 + 40 (-x)^3 + 80 (-x)^2 + 80 (-x) + 32

But the fact that this expansion has been simplified first confuses me in the fact that you can just replace the positive with the negative.

Like you can't do this replacement with a different surd e.g. this is clearly not right

(2-√5)^5 = 362-209√5

So does this only work if the surd changes sign? And is it something "obvious" that doesn't need to be proved because I don't immediately see why it's true.

You can't replace $\sqrt{3}$ with a different surd and quote the same coefficients because the powers of (e.g.) $\sqrt{5}$ will generate different whole number multiples of the surd; what you can do is that if one part of the question asks you to evaluate $(2 + \sqrt{5})^5$ then you can write down the result for $(2 - \sqrt{5})^5$ because the signs of the terms involving the surd will alternate but their magnitude will be the same.

If that was for me, I simply read a 3 as 5. Ill edit the post.

You can't replace $\sqrt{3}$ with a different surd and quote the same coefficients because the powers of (e.g.) $\sqrt{5}$ will generate different whole number multiples of the surd; what you can do is that if one part of the question asks you to evaluate $(2 + \sqrt{5})^5$ then you can write down the result for $(2 - \sqrt{5})^5$ because the signs of the terms involving the surd will alternate but their magnitude will be the same.

Yes I'm aware of that. My point was that if you can just quote (2-√3)^5 = (2+ (-√3))^5 = 362-209√3 without further proof then it could lead to students thinking that e.g. (2-√5)^5 = 362-209√5.

I think my confusion arises from me thinking that this line of working:

(2-√3)^5 = (2+ (-√3))^5 = 362-209√3

was due to the solution writer saying that you can just replace √3 with -√3, but that's not the full story of what's going on.

Now I'm a bit confused - possibly getting a bit old for this You're not quoting that bit in bold "without further proof", you're quoting it as a consequence of what you did in part (a). The point is that your second result is a direct consequence of the first one by just flipping the sign of the surd. Not sure how/why a student would be able to quote the expression 362 - 209root(3) without deriving the initial result, or why they would think that the numbers 362 and 209 would be the same for a different surd, but perhaps I'm just worrying about something that you've already sorted in your own mind

Yes I haven't explained myself very well. To summarise:

(2+√3)^5 = 362+209√3

implying that

(2+ (-√3))^5 = 362-209√3

is not something that is obvious to me and I had to look at the expansion before the simplification to realise where it came from, as explained in the second post.

Normally for these "hence" 1 marker questions I just need to have a think without writing down any working and I can arrive at the answer.

Basically this thread is just me describing my ignorance

Yes I haven't explained myself very well. To summarise:

(2+√3)^5 = 362+209√3

implying that

(2+ (-√3))^5 = 362-209√3

is not something that is obvious to me and I had to look at the expansion before the simplification to realise where it came from, as explained in the second post.

Normally for these "hence" 1 marker questions I just need to have a think without writing down any working and I can arrive at the answer.

Basically this thread is just me describing my ignorance

(2+x)^5 = x^5 + 10 x^4 + 40 x^3 + 80 x^2 + 80 x + 32

It's not obvious "without the expansion", but you did do the expansion to do find $(2+\sqrt{3})^5$.

I see what you're saying. I've just never seen a "hence" question that isn't based solely on the answer of the previous part. Plus the working given in the solution seemed to suggest that it was just based on that.