how to find geometric mean of 1÷3and 64 ÷3

To find the geometric mean of the two numbers \(\frac{1}{3}\) and \(\frac{64}{3}\), you can follow these steps:

### Steps to Find the Geometric Mean:

1. **Multiply the numbers**: 
   \[
   \text{Product} = \left(\frac{1}{3}\right) \times \left(\frac{64}{3}\right)
   \]

2. **Calculate the product**:
   \[
   \text{Product} = \frac{1 \times 64}{3 \times 3} = \frac{64}{9}
   \]

3. **Take the square root** (since we are finding the geometric mean of two numbers):
   \[
   \text{Geometric Mean} = \sqrt{\frac{64}{9}}
   \]

4. **Simplify the square root**:
   \[
   \text{Geometric Mean} = \frac{\sqrt{64}}{\sqrt{9}} = \frac{8}{3}
   \]

So, the geometric mean of \(\frac{1}{3}\) and \(\frac{64}{3}\) is \(\frac{8}{3}\).

Would you like more details or have any questions?

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Here are 5 related questions you might find interesting:

1. How do you find the geometric mean of three numbers?
2. What is the relationship between the geometric mean and arithmetic mean?
3. How can the geometric mean be used in finance or data analysis?
4. What is the geometric mean of two negative numbers?
5. How does the geometric mean differ from the harmonic mean?

**Tip:** The geometric mean is especially useful when dealing with numbers in different ranges or when combining ratios and proportions.

Discover how to find the geometric mean of two fractions, 1/3 and 64/3, with a detailed step-by-step solution. Learn about multiplication of fractions, square roots, and the geometric mean formula. Suitable for high school students.

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