what is the geometric mean of 10 and 30

The geometric mean only applies to positive numbers.

It has a geometric interpretation.  Image a rectangle with sides 10 and 30.  It has an area of 300.  A square of side √(10*30), i.e. one with side lengths equal to the geometric mean of the two rectangle lengths, has the same area.  

Similarly, for higher order geometric means. e.g. a cuboid with side lengths u, v and w has the same volume as a cube with side lengths ³√(u*v*w), etc.

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The geometric mean of N numbers is just the Nth root of their product.....so, since we have two numbers, we have....

 √(10 * 30) = √300 = 10√3

SO

The geometric mean of 7,6,8 and 2

is it   $$\sqrt[4]{7*6*8*2}$$

$${\left({\mathtt{7}}{\mathtt{\,\times\,}}{\mathtt{6}}{\mathtt{\,\times\,}}{\mathtt{8}}{\mathtt{\,\times\,}}{\mathtt{2}}\right)}^{\left({\mathtt{0.25}}\right)} = {\mathtt{5.091\: \!459\: \!790\: \!043\: \!661}}$$

what happen if there is negative numbers?

Thanks Alan