MHB Evaluate a given expression without using a calculator

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The expression $(2^{2^{2014}}+1)(2^{2^{2013}}+1)\cdots(2^{2^2}+1)(2^{2^1}+1)(2^{2^0}+1)+1$ is evaluated without a calculator, leading to a specific numerical result. Participants express gratitude for quick solutions provided in the discussion. New Year wishes are exchanged among members, enhancing the community spirit. The conversation highlights both mathematical evaluation and friendly interactions. Overall, the thread combines problem-solving with festive greetings.
anemone
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Evaluate the expression $(2^{2^{2014}}+1)(2^{2^{2013}}+1)\cdots(2^{2^2}+1)(2^{2^1}+1)(2^{2^0}+1)+1$
 
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anemone said:
Evaluate the expression $(2^{2^{2014}}+1)(2^{2^{2013}}+1)\cdots(2^{2^2}+1)(2^{2^1}+1)(2^{2^0}+1)+1$

wish you happy new year

multiply 1st term by $(2^{2^0}-1) $ which is nothing but 1

and using

$(2^{2^{n-1}}+1)(2^{2^{n-1}}-1) = (2^{2^{n}}-1)$ from n = 1 to 2015 one by one
to get

$(2^{2^{2015}}-1+1)$ or $2^{2^{2015}}$
 
kaliprasad said:
wish you happy new year

multiply 1st term by $(2^{2^0}-1) $ which is nothing but 1

and using

$(2^{2^{n-1}}+1)(2^{2^{n-1}}-1) = (2^{2^{n}}-1)$ from n = 1 to 2015 one by one
to get

$(2^{2^{2015}}-1+1)$ or $2^{2^{2015}}$

Thank you kaliprasad for your quick and nice solution! Well done!

Thanks to you too for the New Year wishes...I too wish you and all a safe New Year filled with peace, joy, robust health, happiness, prosperity and all your dreams come true!;)
 
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