Question

How do you find the geometric mean for the pair of numbers: $-18$ and -36?

Answer 1

Hint: Apply the formula for the geometric mean between two numbers a and b given as G.M = $\pm \sqrt{ab}$. Here G.M denotes the geometric mean. If both the numbers are positive then remove the minus sign and consider only plus sign. In case both the numbers are negative then remove the plus sign and consider only minus sign. This is due to the fact that the geometric mean lies between the two given numbers.

Complete step by step solution:
Here we have been provided with two numbers: - $-18$ and -36 and we are asked to find the geometric mean between them.
Now, we know that the geometric mean between the two numbers a and b is given as G.M = $\pm \sqrt{ab}$. Here G.M denotes the geometric mean. There are two signs plus and minus, however for a particular case we consider only one of them. In case both the numbers are positive then we consider only plus sign and in case both are negative then we consider only minus sign. This is due to the fact that the geometric mean lies between the two numbers.
Let us come to the question. Here we have -18 and -36 and we can see that both of them are negative, so we need to consider only the minus sign.
\[\Rightarrow G.M=-\sqrt{\left( -18 \right)\times \left( -36 \right)}\]
We know that $\left( -1 \right)\times \left( -1 \right)=1$ so we get,
\[\begin{align}
  & \Rightarrow G.M=-\sqrt{18\times 36} \\
 & \therefore G.M=-18\sqrt{2} \\
\end{align}\]
Hence the geometric mean of the given pair of numbers is \[-18\sqrt{2}\].

Note: You may think that what will be the geometric mean if one number is positive and the other is negative. Well the answer is the mean will not be real as the term inside the radical sign will become negative so the mean will become imaginary. In general remember the formula of geometric mean between n numbers is given as \[\sqrt[n]{{{n}_{1}}\times {{n}_{2}}\times {{n}_{3}}\times .......\times {{n}_{n}}}\]