My old calculator only has numbers, ($+$) and ($\times $) keys
Using only those keys, find the last three digits precisely before the decimal point.
$(3 + \sqrt 7)^5$
Note: You can do the calculation without a calculator.
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asked Apr 15 '17 at 7:59
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$\begingroup$ You could have made it a little harder by saying your calculator only has numbers and $+$, since multiplication is repeated addition. $\endgroup$ – Duncan Apr 15 '17 at 15:12
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Clearly $\sqrt 7$ lies between $2.5$ and $3$. Therefore $3 - \sqrt 7 < 1/2$ and the number $(3 - \sqrt 7)^5$ will be very small indeed. We can get rid of the square roots by adding this small number, as follows: $$(3 + \sqrt 7)^5 \\ \approx (3 + \sqrt 7)^5 + (3 - \sqrt 7)^5 \\= 2(3^5 + 10*3^3*7 + 5*3*7^2) \\= 2(243+1890+735) \\= 5736$$
So the actual answer is
$5735$, or just $735$ since you only asked for three digits.
answered Apr 15 '17 at 8:11
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$\begingroup$ you can do slightly better than $\approx$ with $\lt$ $\endgroup$ – JMP Apr 15 '17 at 8:18
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$\begingroup$ Oh right, sorry, misunderstanding on my part. Sorry! $\endgroup$ – TheGreatEscaper Apr 15 '17 at 14:09
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You can determine that:
$2.645751\lt\sqrt{7}\lt2.645752$. Both of these when added to $3$ and raised to the fifth power result in $5735.99\dots$, so the answer is $735$.
