Question

How do you find the geometric mean between $18$ and $1.5$

Answer 1

Hint: Given the numbers and the geometric mean between them is determined by multiplying the provided numbers and finding the nth root of the product of the numbers.

Formula used:
\[GM = \sqrt[n]{{ab}}\]
Here, $a$ and $b$ are the numbers and GM is the geometric mean between these numbers

Complete step-by-step answer:
Given two numbers $a$ and $b$ in which the numbers are in geometric sequence $a,x,b$ where $x$ is the geometric mean of two numbers.
Now apply the definition of the geometric mean by substituting \[a = 18\] and \[b = 1.5\]
\[ \Rightarrow \dfrac{{18}}{x} = \dfrac{x}{{1.5}}\]
Now we will cross multiply the terms.
\[ \Rightarrow {x^2} = 18 \times 1.5\]
Now we will multiply the two numbers.
\[ \Rightarrow {x^2} = 27\]
Now we will find the square root of the resultant number as the value of \[n\]is \[2\].
\[ \Rightarrow x = \sqrt {27} \]
Now, we will find the factors of the number.
\[ \Rightarrow x = \sqrt {3 \times 3 \times 3} \]
Now we will apply the product property of square root.
\[ \Rightarrow x = 3\sqrt 3 \]

Final answer: Hence the geometric mean between $18$ and $1.5$ is \[3\sqrt 3 \]

Additional information: In the geometric mean the middle term of the series containing three numbers which are in G.P. is known as the geometric mean between the two numbers. The geometric mean is determined by taking the positive values of the data set. The geometric mean of the numbers is used to define the central tendency of the set of values.

Note:
In such types of questions students mainly get confused in applying the formula. As they don't know which formula they have to apply. So when the geometric mean between two numbers is given then apply the relation between the two numbers which is equal to the geometric mean between them.