Question

find the geometric mean of 8 and 28.
the geometric mean is \boxed{}.

Explanation:

Step1: Recall the formula for geometric mean

The formula for the geometric mean of two numbers \(a\) and \(b\) is \(\sqrt{ab}\). Here \(a = 8\) and \(b=28\).

Step2: Calculate the product \(ab\)

\(ab=8\times28 = 224\).

Step3: Simplify the square - root

\(\sqrt{224}=\sqrt{16\times14}\). Using the property \(\sqrt{mn}=\sqrt{m}\times\sqrt{n}\) (\(m = 16\), \(n = 14\)), we get \(\sqrt{16}\times\sqrt{14}=4\sqrt{14}\approx4\times3.742 = 14.968\approx15\) (rounded to the nearest whole number). Also, \(\sqrt{224}=\sqrt{4\times56}=2\sqrt{56}=2\sqrt{4\times14}=4\sqrt{14}\). If we calculate \(\sqrt{224}\) exactly, \(\sqrt{224}=\sqrt{16\times14}=4\sqrt{14}\approx14.966\).

Answer:

\(4\sqrt{14}\approx15\)