1. **State the problem:** Find the geometric mean between 5 and 10. 2. **Formula:** The geometric mean $G$ between two numbers $a$ and $b$ is given by: $$G = \sqrt{a \times b}$$ 3. **Apply the formula:** $$G = \sqrt{5 \times 10}$$ 4. **Calculate the product:** $$5 \times 10 = 50$$ 5. **Calculate the square root:** $$G = \sqrt{50}$$ 6. **Simplify the square root:** $$\sqrt{50} = \sqrt{25 \times 2} = \sqrt{25} \times \sqrt{2} = 5\sqrt{2}$$ 7. **Final answer:** The geometric mean between 5 and 10 is $5\sqrt{2}$.