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ololo11 [35]
4 years ago
12

What is the geometric mean of 6 and 16?

Mathematics
1 answer:
Ann [662]4 years ago
8 0

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}                                                                                        

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