# Find last 3 digits before decimal point

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.

• You could have made it a little harder by saying your calculator only has numbers and $+$, since multiplication is repeated addition. – Duncan Apr 15 '17 at 15:12

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: $$(3 + \sqrt 7)^5 \\ \approx (3 + \sqrt 7)^5 + (3 - \sqrt 7)^5 \\= 2(3^5 + 10*3^3*7 + 5*3*7^2) \\= 2(243+1890+735) \\= 5736$$
$5735$, or just $735$ since you only asked for three digits.
• you can do slightly better than $\approx$ with $\lt$ – JMP Apr 15 '17 at 8:18
$2.645751\lt\sqrt{7}\lt2.645752$. Both of these when added to $3$ and raised to the fifth power result in $5735.99\dots$, so the answer is $735$.