ind the geometric mean of each pair of numbers. If necessary, give the answer in simples dical form. \( \begin{array}{ll}3 \text { and } 27 & \text { 2. } 5 \text { and } 15\end{array} \)

To find the geometric mean of a pair of numbers, you take the square root of the product of the two numbers. For 3 and 27: The geometric mean is \(\sqrt{3 \times 27} = \sqrt{81} = 9\). For 5 and 15: The geometric mean is \(\sqrt{5 \times 15} = \sqrt{75} = \sqrt{25 \times 3} = 5\sqrt{3}\). So the geometric means are 9 and \(5\sqrt{3}\). Now, let's have some fun with this mathematical concept! The geometric mean is often used in fields such as finance or biology; for example, when averaging growth rates over time, it provides a more accurate representation than simple averages. And here’s a little tip: when working with geometric means, remember that they can only be computed with non-negative numbers! A common mistake is to try and find the geometric mean of a negative pair, which simply doesn't work in the realm of real numbers!