I see that you've uploaded an image containing several geometric mean problems. I'll solve each of them step by step.

1. Find the geometric mean between 9 and 324.

The geometric mean $G$ between two numbers $a$ and $b$ is calculated using the formula: $G = \sqrt{a \times b}$ Substituting the values: $G = \sqrt{9 \times 324} = \sqrt{2916} = 54$ So, the geometric mean between 9 and 324 is 54.

2. What are the two geometric means between 3 and 648?

When finding two geometric means between two numbers $a$ and $b$, we assume they form a geometric sequence: $a, G_1, G_2, b$. The common ratio $r$ can be found by: $r = \sqrt[3]{\frac{b}{a}}$ Substituting the values: [ r = \sqrt[3]{\frac{648}{3}} = \sqrt[3]{216} = 6 ] Now, calculate the two geometric means: $G_1 = a \times r = 3 \times 6 = 18$ $G_2 = G_1 \times r = 18 \times 6 = 108$ So, the two geometric means are 18 and 108.

3. Find the three geometric means between 1536 and 6.

For three geometric means $G_1, G_2, G_3$, we have: $r = \sqrt[4]{\frac{b}{a}}$ Substituting the values: [ r = \sqrt[4]{\frac{6}{1536}} = \sqrt[4]{\frac{1}{256}} = \sqrt[4]{\frac{1}{2^8}} = \frac{1}{4} ] Now calculate the three geometric means: $G_1 = 1536 \times \frac{1}{4} = 384$ $G_2 = 384 \times \frac{1}{4} = 96$ $G_3 = 96 \times \frac{1}{4} = 24$ So, the three geometric means are 384, 96, and 24.

4. Give 3 geometric means between 840 and 15.

Similarly, for three geometric means: [ r = \sqrt[4]{\frac{15}{840}} = \sqrt[4]{\frac{1}{56}} \approx \sqrt[4]{0.017857} \approx 0.34 ] Now calculate: $G_1 = 840 \times 0.34 \approx 285.6$ $G_2 = 285.6 \times 0.34 \approx 97.1$ $G_3 = 97.1 \times 0.34 \approx 33$ So, the three geometric means are approximately 285.6, 97.1, and 33.

5. What are the 2 geometric means between 5 and 320?

For two geometric means: [ r = \sqrt[3]{\frac{320}{5}} = \sqrt[3]{64} = 4 ] Now calculate: $G_1 = 5 \times 4 = 20$ $G_2 = 20 \times 4 = 80$ So, the two geometric means are 20 and 80.

Would you like further details or explanations? Here are five related questions for practice:

1. What is the geometric mean between 8 and 200?
2. Find three geometric means between 12 and 96.
3. Calculate the geometric mean between 45 and 5.
4. Determine the four geometric means between 1024 and 16.
5. What are the two geometric means between 7 and 448?

Tip: When finding multiple geometric means between two numbers, always remember that the numbers form a geometric sequence. The number of geometric means corresponds to the number of divisions you need to make in the ratio.