How to Calculate 3^2048: Step-by-Step Guide

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In summary, to prove the given equation, start by using the given answer of 3^2048. From there, use the formula a^2 - b^2 = (a-b)(a+b) with a = 2^2048 and b = 3^2048. Repeat this process with a = 2^1024 and b = 3^1024, and continue until the last pair on the left. This will result in (2-3)(2+3), giving a minus sign in the equation.
  • #1
tsuwal
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the answer is 3^2048. How do I get there?
 
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  • #2
Since you are given the answer, use that information!

You have to prove that
##(2+3)(2^2+3^2)\cdots(2^{2048} + 3^{2048}) + 2^{4096} - 3^{4096} = 0##
Now, think what you can do with ##2^{4096} - 3^{4096}## ...
 
  • #3
i don't know, what can i do :S?
 
  • #4
and the answer is not given, it's multiple choice
 
  • #5
a2 - b2 = (a-b)(a+b)

Start with a = 22048 and b = 32048
next repeat with a = 21024 and b = 31024
etc.
At the end you will have (2-3)(2+3). Just be careful with the sign.
 
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  • #6
but i got a plus sign not a minus sign...
 
  • #7
tsuwal said:
but i got a plus sign not a minus sign...
No, its a minus sign:
##(2+3)(2^2+3^2)\cdots(2^{2048} + 3^{2048}) + 2^{4096} - 3^{4096} = 0##
AlephZero was referring to the last pair on the left.
 
  • #8
now i get it. thanks!
 

Related to How to Calculate 3^2048: Step-by-Step Guide

1. How do I calculate 3^2048?

To calculate 3^2048, you can use a calculator or a computer program. Simply enter "3^2048" into the calculator or program and it will give you the answer.

2. Can I calculate 3^2048 by hand?

Technically, yes, you can calculate 3^2048 by hand. However, it would be extremely time-consuming and prone to error. It is highly recommended to use a calculator or computer program for such large calculations.

3. What is the result of 3^2048?

The result of 3^2048 is a very large number, approximately 1.42 x 10^978. This is because 3 multiplied by itself 2048 times results in a very large number with 978 digits.

4. Can I simplify the calculation of 3^2048?

No, 3^2048 cannot be simplified any further. It is already in its simplest form.

5. Are there any tips for calculating 3^2048?

If you are using a calculator or computer program, make sure to double check your input to avoid any errors. It may also be helpful to break the calculation into smaller parts, such as calculating 3^1024 twice and then multiplying the results together.

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