Geometric Mean Calculation for Numbers $2, 8, 18, 27$

This guide provides a step-by-step explanation to calculate the geometric mean for the given set of numbers: $2$, $8$, $18$, and $27$. We will cover the definition of geometric mean and demonstrate the calculation process.

Understanding the Geometric Mean Concept

The geometric mean is a useful measure of central tendency, especially when dealing with values that multiply or grow exponentially. Unlike the arithmetic mean (which sums values and divides by the count), the geometric mean uses the product of the values. For a set of $n$ non-negative numbers $\{x_1, x_2, \dots, x_n\}$, the geometric mean (GM) is defined as the $n$-th root of the product of these numbers.

The mathematical formula is:

$$ \text{GM} = \sqrt[n]{x_1 \times x_2 \times \dots \times x_n} $$

In this problem, we have $n=4$ numbers.

Calculating the Geometric Mean Step-by-Step

We need to find the geometric mean of the numbers $2$, $8$, $18$, and $27$. Let's apply the formula.

Step 1: Identify the Numbers and Count

The numbers are $x_1 = 2$, $x_2 = 8$, $x_3 = 18$, $x_4 = 27$. The total count of numbers is $n=4$.

Step 2: Compute the Product of the Numbers

Multiply all the numbers together:

Product $= 2 \times 8 \times 18 \times 27$

Calculate the product step-wise:

• $2 \times 8 = 16$
• $16 \times 18 = 288$
• $288 \times 27 = 7776$

So, the product of the numbers is $7776$.

Step 3: Calculate the n-th Root of the Product

Now, we need to find the 4th root of the product $7776$, since $n=4$.

$$ \text{GM} = \sqrt[4]{7776} $$

To simplify the radical, we find the prime factorization of $7776$:

• $7776 = 2 \times 3888$
• $7776 = 2^2 \times 1944$
• $7776 = 2^3 \times 972$
• $7776 = 2^4 \times 486$
• $7776 = 2^5 \times 243$
• Since $243 = 3^5$, we have $7776 = 2^5 \times 3^5$.

We can rewrite the product using its prime factors:

$7776 = 2^5 \times 3^5 = (2 \times 3)^5 = 6^5$

Substitute this back into the geometric mean formula:

$$ \text{GM} = \sqrt[4]{6^5} $$

To simplify $\sqrt[4]{6^5}$, we can express $6^5$ as $6^4 \times 6^1$.

$$ \text{GM} = \sqrt[4]{6^4 \times 6^1} $$

Using the property $\sqrt[n]{a \times b} = \sqrt[n]{a} \times \sqrt[n]{b}$, we get:

$$ \text{GM} = \sqrt[4]{6^4} \times \sqrt[4]{6^1} $$

Since $\sqrt[4]{6^4} = 6$, the expression simplifies to:

$$ \text{GM} = 6 \times \sqrt[4]{6} $$

$$ \text{GM} = 6\sqrt[4]{6} $$

Final Result

The geometric mean of the numbers $2$, $8$, $18$, and $27$ is $6\sqrt[4]{6}$.