Express the number one million using numbers containing only the digit 9 and algebraic operations of addition, subtraction, multiplication, division, exponentiation, and root. Find at least three different solutions.

(L. Dedková)

Hint. Express as many small natural numbers as possible in the given way, which could be useful further on.
Natural numbers containing only the digit 9 are 9, 99, 999, 9999, etc. Random operations with these numbers yield various results, but we can notice, for example, the following results:

$$
\frac{9}{9}=1, \quad \sqrt{9}=3, \quad 9+9=18, \quad 99-9=90 \quad \text { etc. }
$$

In the next step, we can express the number 10, for example, as follows:

$$
10=9+\frac{9}{9}=\frac{9 \cdot 9+9}{9}=\frac{99-9}{9}
$$

Similarly, we can express 100, 1000, etc., and even a million:

$$
1000000=999999+\frac{9}{9}=\frac{999999 \cdot 9+9}{9}=\frac{9999999-999999}{9} .
$$

From other ideas in the first step, we can express, for example,

$$
2=\frac{9}{9}+\frac{9}{9}=\frac{9+9}{9}, \quad 6=9-\sqrt{9}=\frac{9+9}{\sqrt{9}} \quad \text { etc. }
$$

From this, a million can be expressed in many other ways, for example, as follows:

$$
1000000=\left(9+\frac{9}{9}\right)^{9-\sqrt{9}}
$$

Note. Using $\frac{9}{9}=1$, a million can also be expressed as the sum of a million of these fractions. However, this and similar ideas cannot be evaluated if they are not implemented in the manner described above (i.e., without words or dots indicating the continuation of a certain idea).