Question

Can a geometric mean be negative?

Answer 1

Hint: We'll need to look at a few examples to arrive at the conclusion. We'll use two examples to demonstrate this. One is made up of positive numbers, while the other is made up of negative numbers. After that, we'll apply a formula, \[{{({{X}_{1}}.{{X}_{2}}.{{X}_{3}})}^{\dfrac{1}{n}}}\,\,or,\sqrt[\dfrac{1}{n}]{{{X}_{1}}.{{X}_{2}}.{{X}_{3}}}\]and the cases that pass through it will be the correct ones.

Complete step by step answer:
To comprehend its notion, we must first define the phrase geometric mean. Simply put, the geometric is the nth root of the words multiplied together. Consider the terms and multiply them together as an example. As a result, we have. Now we'll take the nth root of it. \[{{({{X}_{1}},{{X}_{2}},{{X}_{3}})}^{\dfrac{1}{n}}}\,\,or,\sqrt[\dfrac{1}{n}]{{{X}_{1}}.{{X}_{2}}.{{X}_{3}}}\]. To grasp the concept of whether the geometric mean can be negative or not, we must first explore two scenarios. We'll take all the positive values in this first scenario. Take a look at the numbers 4, 2, and 3. Because there are only three numbers, so\[n=3\]. Let us now calculate its geometric mean. According to the formula\[{{({{X}_{1}}.{{X}_{2}}.{{X}_{3}})}^{\dfrac{1}{n}}}\]. The geometric mean of these numbers will be as\[\,\,{{(4.2.3)}^{\dfrac{1}{3}}}\]. As a result, we obtain, \[\,\,{{(12)}^{\dfrac{1}{3}}}=2.289\] and it's evident that it's a positive number.
We'll now look at the case of negative numbers. Let's pretend we're trying to figure out what the geometric mean of the numbers -4,-2,-3 is. As a result\[\,\,{{(-4.-2.-3)}^{\dfrac{1}{3}}}=\,\,{{(-12)}^{\dfrac{1}{3}}}\]. The cube root of a negative number cannot be found in this situation. There isn't a way to do it. As a result, the geometric mean of -4,-2,-3 cannot be found. As a result, a geometric mean cannot be negative because the geometric mean formula can only be applied to positive values.

Note: It's important to remember that we can't determine the geometric mean of a negative number, and we can't calculate the square root of a negative number, either. As a result, the square root of a negative number, as well as the cube roots, are not defined. However, if there are an even number of negative values, we can calculate the geometric mean. Because if the numbers are multiplied together, they will automatically become positive. This is why, before answering any negative question, we should examine the numbers to see if they are even or not. For example, the numbers we selected for negative integers in this question were strange. We could have estimated its geometric mean if we only took -4 and -2.