# How do you find the square root of 85?

Mar 26, 2015

$\sqrt{85}$ is a non-terminating, non-repeating decimal. You can't finish finding it.
But you can get better and better approximations by: start with a number whose square is close to 85. (9 or 10. ${9}^{2}$ is closer, so start there.

Now divide 85 by $9$ to get $9.4444$ (to 4 places)
Average the two numbers $9$ and $9.4444$
$\frac{9 + 9.4444}{2} = 9.2222$

Repeat:
$85 \div 9.2222 = 9.2169$ (rounded to 4 places)
Average: $\frac{9.2222 + 9.2169}{2} = 9.2196$

Repeat:
$85 \div 9.2196 = 9.2195$ (rounded to 4 places)
Average: $\frac{9.2196 + 9.2195}{2} = 9.2196$

$\sqrt{85} \approx 9.2196$

That answer is accurate to 4 decimal places. If you need more accuracy, keep more decimal places at each step.