To find two geometric means between 7 and 189, we can follow the process of inserting two numbers in a geometric sequence. Let the four terms of the sequence be:

$7, G_1, G_2, 189$

Since the numbers form a geometric sequence, the ratio between consecutive terms must be constant. Let that ratio be $r$.

Thus, we have the following relations: $G_1 = 7r, \quad G_2 = 7r^2, \quad 189 = 7r^3$

Now, solve for $r$: $189 = 7r^3 \implies r^3 = \frac{189}{7} = 27 \implies r = \sqrt[3]{27} = 3$

Now that we know $r = 3$, we can find the two geometric means:

$G_1 = 7r = 7 \times 3 = 21$ $G_2 = 7r^2 = 7 \times 3^2 = 7 \times 9 = 63$

Thus, the two geometric means between 7 and 189 are 21 and 63.

Would you like further details or have any questions?

Here are five related questions you might find interesting:

1. How do you find three geometric means between two numbers?
2. What is the formula for the $n$-th term of a geometric sequence?
3. How do arithmetic means differ from geometric means?
4. How do you determine the common ratio if three terms are given in a geometric sequence?
5. Can geometric means be negative?

Tip: In a geometric sequence, the ratio between consecutive terms is constant, and multiplying the first term by the common ratio raised to different powers gives the terms.