To find the geometric mean between two numbers, we use the following formula:

\[
\text{Geometric mean} = \sqrt{a \times b}
\]

where \(a\) and \(b\) are the two numbers.

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

Now, calculate the geometric mean:

\[
\text{Geometric mean} = \sqrt{\frac{1}{3} \times 64} = \sqrt{\frac{64}{3}}
\]

Thus, the geometric mean is:

\[
\text{Geometric mean} = \sqrt{\frac{64}{3}} \approx 4.6188
\]

Let me know if you want more details or have any questions.

Here are some related questions for practice:

1. What is the geometric mean of 5 and 20?
2. Find the geometric mean of \(\frac{2}{5}\) and \(\frac{25}{2}\).
3. Calculate the geometric mean of 9 and 16.
4. How is the geometric mean different from the arithmetic mean?
5. What is the geometric mean of 10 and 0.1?

**Tip:** The geometric mean is most useful when comparing quantities that multiply together, such as growth rates.

Learn how to calculate the geometric mean between 1/3 and 64 using the formula sqrt(a * b). Understand the steps involved in finding the geometric mean of two numbers and explore related mathematical concepts like square roots. Suitable for students in Grades 7-9.

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