The geometric mean of two numbers $a$ and $b$ is given by the formula:

$\text{Geometric Mean} = \sqrt{a \times b}$

For the numbers $\frac{2}{3}$ and $6$, we substitute these values into the formula:

$\text{Geometric Mean} = \sqrt{\frac{2}{3} \times 6}$

Now, calculate the product inside the square root:

$\frac{2}{3} \times 6 = \frac{2 \times 6}{3} = \frac{12}{3} = 4$

So, the geometric mean is:

$\text{Geometric Mean} = \sqrt{4} = 2$

Conclusion

The geometric mean between $\frac{2}{3}$ and $6$ is 2.

Would you like more details or have any questions?

Related Questions:

1. What is the geometric mean of three or more numbers?
2. How do you calculate the geometric mean for a set of negative numbers?
3. How is the geometric mean used in finance and statistics?
4. What is the difference between the arithmetic mean and the geometric mean?
5. How can the geometric mean be applied in geometric progression?

Tip:

The geometric mean is especially useful in cases involving growth rates and percentages because it accurately reflects the compounding effect over time.