Find the geometric mean between 1/3 and 64/3

The geometric mean between two numbers \( a \) and \( b \) is given by the formula:

\[
\text{Geometric Mean} = \sqrt{a \cdot b}
\]

In this case, \( a = \frac{1}{3} \) and \( b = \frac{64}{3} \).

Let's calculate the geometric mean:

\[
\text{Geometric Mean} = \sqrt{\frac{1}{3} \cdot \frac{64}{3}} = \sqrt{\frac{1 \cdot 64}{3 \cdot 3}} = \sqrt{\frac{64}{9}} = \frac{\sqrt{64}}{\sqrt{9}} = \frac{8}{3}
\]

So, the geometric mean between \( \frac{1}{3} \) and \( \frac{64}{3} \) is:

\[
\frac{8}{3}
\]

Would you like more details on this, or do you have any questions?

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Here are 5 related questions:
1. How do you find the geometric mean between three numbers?
2. What is the difference between arithmetic and geometric means?
3. Can the geometric mean be negative?
4. How is the geometric mean used in real-world applications?
5. What is the geometric mean of 4 and 16?

**Tip**: When dealing with fractions, multiplying and simplifying under the square root is a useful technique to avoid mistakes.

Find the geometric mean between 1/3 and 64/3.

Learn how to find the geometric mean between the fractions 1/3 and 64/3 using the geometric mean formula. This step-by-step solution breaks down the multiplication and square root process, providing a clear explanation for students in Grades 6-8.

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