Question

What is the geometric mean of 8 and 18?

Answer 1

Hint: For solving this question you should know about the geometric mean of any numbers. The geometric mean of any number is equal to the under root or square root of multiplication of both of that number and the ratio of both the numbers with the ratio with each number and geometric mean is always equal to each other. If we consider two positive numbers a & b then the geometric mean of this is \[\sqrt{a.b}\] and the ratio \[a:\sqrt{ab}\] is equal to the ratio \[\sqrt{ab}:b\].

Complete step by step answer:
According to the question it is clear that we have to calculate the geometric mean of two positive numbers 8 and 18.
The geometric mean of two positive numbers a and c is the positive number b in such a way that it satisfies the condition \[\dfrac{a}{b}=\dfrac{b}{c}\].
The geometric mean of a and b is \[\sqrt{ab}\] if a and b are both positive numbers and it will be \[-\sqrt{ab}\] if a and b are negative numbers.
The geometric mean is the positive square root of the product of two numbers.
For a square the geometric mean of two numbers a and b, is the length of one side of a square whose area is equal to the area of a rectangle with the sides of a and b lengths.
If we talk about the n numbers then the geometric mean is the \[{{n}^{th}}\] root of the product of n numbers. If n positive values are given then the geometric mean is the \[{{n}^{th}}\] positive root of the product of that n – numbers.
So, the geometric mean is equal to \[\sqrt{a.b}\]
Here, \[a=8\]
and \[b=18\]
The geometric mean of this is equal to \[=\sqrt{8.18}=\sqrt{144}\]
Further we can write it as: \[\sqrt{144}=12\]

The geometric mean of 8 and 18 is equal to 12.

Note: The geometric mean is generally a \[{{n}^{th}}\] root of n positive numbers. And it is defined as the (product of n - number) \[^{\dfrac{1}{n}}\]. To calculate the geometric mean of ‘n’ numbers you should multiply them with one another completely and then make it as a power \[\dfrac{1}{n}\].