Question

The geometric mean of $2,6,8,24$ is
$
  A.\sqrt {48} = 4\sqrt 3 \\
  B.166 \\
  C.24 \\
  D.36 \\
$

Answer 1

Hint: In this question to calculate the geometric mean of the given numbers, we use the following formula i.e. $\sqrt[n]{{{a_1} \times {a_2} \times {a_3}........{a_n}}}$.

Complete step-by-step answer:
According to the question the series is given i.e. $2,6,8,24$ and we have to find the geometric mean of it.
We know that the Geometric Mean of $n$ terms is \[{\left( {Product{\text{ }}of{\text{ }}terms} \right)^{\dfrac{1}{n}}}\]
$n = 4$ because there are four terms in the series.
$\therefore $ Geometric Mean $ = {\left( {2 \times 6 \times 8 \times 24} \right)^{\dfrac{1}{4}}}$
$ = {\left( {{2^8} \times {3^2}} \right)^{\dfrac{1}{4}}} = {2^2} \times {3^{\dfrac{1}{2}}}$
$ = 4\sqrt 3 = \sqrt {48} $
Hence the geometric mean of $2,6,8,24$ is $ = 4\sqrt 3 = \sqrt {48} $

Note: It is recommended to remember these formulas while involving geometric mean questions, as it saves a lot of time. Eventually it will be difficult but with practice things get easier.