- #1
tsuwal
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the answer is 3^2048. How do I get there?
How to Calculate 3^2048: Step-by-Step Guide
- Context: High School
- Thread starter tsuwal
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Discussion Overview
The discussion revolves around the calculation of \(3^{2048}\) and the methods to derive it through algebraic manipulation and identities. Participants explore different approaches to prove a specific equation related to the powers of 2 and 3.
Discussion Character
- Mathematical reasoning
- Homework-related
- Debate/contested
Main Points Raised
- One participant states that the answer is \(3^{2048}\) and seeks guidance on how to arrive at that conclusion.
- Another participant suggests proving the equation \((2+3)(2^2+3^2)\cdots(2^{2048} + 3^{2048}) + 2^{4096} - 3^{4096} = 0\) as a method to find the answer.
- A participant expresses uncertainty about how to proceed with the suggested proof.
- One participant clarifies that the answer is not given but is part of a multiple-choice question.
- Another participant introduces the difference of squares formula, proposing to start with specific values for \(a\) and \(b\) and iteratively apply the formula.
- There is a discussion about the signs in the equation, with one participant noting a discrepancy between a plus and minus sign in their calculations.
- Clarification is provided regarding the signs in the equation, leading to a participant expressing understanding after the explanation.
Areas of Agreement / Disagreement
Participants do not reach a consensus on the method to calculate \(3^{2048}\), as there are differing interpretations of the signs in the equation and the steps to take in the proof.
Contextual Notes
There are unresolved aspects regarding the manipulation of the equation and the assumptions made about the signs in the calculations. The discussion relies on specific algebraic identities that may not be universally agreed upon.
the answer is 3^2048. How do I get there?
- #2
AlephZero
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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}## ...
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
tsuwal
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i don't know, what can i do :S?
- #4
tsuwal
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and the answer is not given, it's multiple choice
- #5
mathman
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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.
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.
Last edited:
- #6
tsuwal
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but i got a plus sign not a minus sign...
- #7
HallsofIvy
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No, its a minus sign:tsuwal said:
##(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
tsuwal
- 105
- 0
now i get it. thanks!
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