1. **State the problem:** Find the geometric mean between the pairs of numbers given: I. 9 and 16 II. - \frac{3}{10} and - \frac{5}{6} 2. **Recall the formula for geometric mean:** The geometric mean of two numbers $a$ and $b$ is given by: $$ \text{Geometric Mean} = \sqrt{a \times b} $$ 3. **Calculate for pair I (9 and 16):** $$ \sqrt{9 \times 16} = \sqrt{144} = 12 $$ The geometric mean of 9 and 16 is 12. 4. **Calculate for pair II (-\frac{3}{10} and -\frac{5}{6}):** First multiply the numbers: $$ -\frac{3}{10} \times -\frac{5}{6} = \frac{3 \times 5}{10 \times 6} = \frac{15}{60} = \frac{1}{4} $$ Now take the square root: $$ \sqrt{\frac{1}{4}} = \frac{1}{2} $$ The geometric mean of -\frac{3}{10} and -\frac{5}{6} is \frac{1}{2}. **Note:** The geometric mean for pair II is positive since the product of two negative numbers is positive. **Final answers:** - Geometric mean of 9 and 16 is $12$. - Geometric mean of -\frac{3}{10} and -\frac{5}{6} is $\frac{1}{2}$.