Expansion of expressions: Some instances of the expression (𝐴+𝐵√𝐶)^𝐷 , in which 𝐴,𝐵,𝐶, and 𝐷 represent distinct single non-zero digits, may...

[paulrwoolley]

Hi
This is the preamble to a numerical puzzle I have been set:

Some instances of the expression (𝐴+𝐵√𝐶)^𝐷 , in which 𝐴,𝐵,𝐶, and 𝐷 represent distinct single non-zero digits, may be expanded to give 𝐸+𝐹√𝐶 such that E is a four-digit number, also containing distinct non-zero digits.

Can anyone deduce anything from this, or perhaps simplify in such a way that if one were given 𝐸, the possibilities for 𝐴,𝐵,𝐶 and 𝐷 could be calculated.
Equally, could anyone explain how to obtain possible answers for E when given A,B,C and D?

Thanks in advance.

 

[Steven G]

I think that you should read the posting guidelines if you want help with your problem.

 

[paulrwoolley]

I think that you should read the posting guidelines if you want help with your problem.

Hi Steven
Please can you elaborate?
Thanks
Paul

 

[Dr.Peterson]

Some instances of the expression (𝐴+𝐵√𝐶)^𝐷 , in which 𝐴,𝐵,𝐶, and 𝐷 represent distinct single non-zero digits, may be expanded to give 𝐸+𝐹√𝐶 such that E is a four-digit number, also containing distinct non-zero digits.

Can anyone deduce anything from this, or perhaps simplify in such a way that if one were given 𝐸, the possibilities for 𝐴,𝐵,𝐶 and 𝐷 could be calculated.
Equally, could anyone explain how to obtain possible answers for E when given A,B,C and D?

I'd first try some examples, such as expanding [imath]\left(1+2\sqrt{3}\right)^2[/imath] and [imath]\left(1+2\sqrt{3}\right)^4[/imath]. What do you find? Does that suggest anything more general?

But I'm not sure I understand that part about digits (which digits have to be distinct?); in any case, there will probably be a lot of trial and error (that is, "brute force"), not a simple formula.

What is the entire puzzle, and where does it come from?