Alg2.1.27

Find the geometric mean of $8+2\sqrt{15},11-2\sqrt{30}\,$

Geometric mean of two numbers is the square root of their product

Finding individual square roots

$\sqrt{8+2\sqrt{15}}=\sqrt{8+2\sqrt{5\cdot3}}\,$

$\sqrt{8+2\sqrt{15}}=\sqrt{(5+3+2\sqrt{5\cdot3}}\,$

$\sqrt{(\sqrt{5})^2+(\sqrt{3})^2+2\sqrt{5\cdot3}}\,$

$\sqrt{(\sqrt{5}+\sqrt{3})^2}\,$

The square root of the first one is

$\sqrt{5}+\sqrt{3}\,$

Similarly the square root for the second given number is

$\sqrt{11-2\sqrt{30}}=\sqrt{6+5-2\sqrt{6\cdot5}}\,$

$\sqrt{11-2\sqrt{30}}=\sqrt{(6+5-2\sqrt{6\cdot5}}\,$

$\sqrt{(\sqrt{6})^2+(\sqrt{5})^2+2\sqrt{6\cdot5}}\,$

$\sqrt{(\sqrt{6}-\sqrt{5})^2}\,$

The square root of the second one is one is

$\sqrt{6}-\sqrt{5}\,$

Now the geometric mean is

$(\sqrt{5}+\sqrt{3})(\sqrt{6}-\sqrt{5})\,$