# How can I write the numbers 5 and 7 as some sequence of operations on three 9s?

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?