Question

What is the geometric mean of 6 and 16?

Answer 1

Answer:

The geometric mean of 6 and 16 = $4\sqrt{6}$

Step-by-step explanation:

The Geometric Mean is basically a special type of average which can be determined

• by multiply the numbers together and subsequently taking a square root in case there are two numbers; but we take cube root in case there three numbers etc.

So, lets take the geometric mean of 6 and 16

Considering the number

6 and 16

• Step 1: Multiply the numbers i.e.  6 × 16 = 96

Step 2: As there are two number, so just take the square root i.e. $\sqrt{96} =4\sqrt{6}$

In other words:

Geometric Mean of 6 and 16:

                                                  $\sqrt{6\times16}$

                                                  $=\sqrt{96}$                                        

  $\mathrm{Prime\:factorization\:of\:}96:\quad 2^5\cdot \:3$

                                                  $=\sqrt{2^5\cdot \:3}$

                                                  $=\sqrt{2^4\cdot \:2\cdot \:3}$

$\mathrm{Apply\:radical\:rule}:\quad \sqrt{ab}=\sqrt{a}\sqrt{b},\:\quad \:a\ge 0,\:b\ge 0$

                                                   $=\sqrt{2^4}\sqrt{2\cdot \:3}$

$\mathrm{Apply\:radical\:rule}:\quad \sqrt[n]{a^m}=a^{\frac{m}{n}}$      

                                                    $=2^{\frac{4}{2}}\sqrt{2\cdot \:3}$

                                                    $=2^2\sqrt{2\cdot \:3}$

                                                     $=4\sqrt{6}$

Therefore, the geometric mean of 6 and 16 = $4\sqrt{6}$