What is the geometric mean of 6, 8, 16 and 27?

Concept:

The geometric mean is defined as the nth root of the product of n numbers.

The geometric mean of a data set \(\rm {\textstyle \left\{a_{1},a_{2},\,\ldots ,\,a_{n}\right\}}\) is given by: 

\(\rm GM = \rm {\textstyle \left\{a_{1}\times a_{2}\times\,\ldots \times\,a_{n}\right\}}^{\frac{1}{n}}\)

Calculation:

To Find: Geometric mean of 6, 8, 16 and 27

Here n = 4

Now,

 \(\rm GM = \rm ({{6 \times 8 \times 16 \times 27}})^{\frac{1}{4}}\)

\(= \rm ({{2 \times 3 \times 2^3 \times 2^4 \times 3^3}})^{\frac{1}{4}} \\ =(2^8 \times 3^4)^{\frac{1}{4}}\\ = (2^2 \times 3)\\=12\)