TITLE: How can I write the numbers 5 and 7 as some sequence of operations on three 9s?
QUESTION [47 upvotes]: I want to make the numbers $1, 2, ..., 9$ using exactly three copies of the number $9$ and the following actions: addition, subtraction, multiplication, division, squaring, taking square roots, and other action.
How can we make the numbers $5$ or $7$?
For example, we can make the below numbers using exactly three copies of the number 9.

$1=\dfrac{\sqrt 9\times\sqrt9}{9}$
$2=\dfrac{9+9}{9}$
$3=\dfrac{\sqrt9\times9}{9}$
$4=\dfrac{9}{9}+\sqrt9$
$5=\,?$
$6=\dfrac{9+9}{\sqrt9}$
$7=\,?$
$8=9-\dfrac{9}{9}$
$9=9+9-9.$

Now, how can we make the numbers 5 and 7?

REPLY [3 votes]: Try $\lceil\sqrt[9]{9!}\rceil$. If you insist on three 9s try $\lfloor\sqrt[9]{9!}\rfloor+0.\bar 9$.