Geometric Mean Problem: Solve for x

This solution explains how to find the value of $x$ when the geometric mean of 2, 6, and $x$ is given as 6.

Geometric Mean Formula

The geometric mean (GM) of $n$ numbers is calculated by taking the $n$-th root of the product of the numbers. For three numbers ($a, b, c$), the formula is:

$ GM = \sqrt[3]{a \cdot b \cdot c} $

Applying Geometric Mean to Problem

We are given:

• Numbers: 2, 6, $x$
• Geometric Mean: 6

Substitute these values into the formula:

$ \sqrt[3]{2 \cdot 6 \cdot x} = 6 $

Solving for x

To find the value of $x$, follow these steps:

1. Simplify the expression under the cube root: $ \sqrt[3]{12x} = 6 $
2. Cube both sides to remove the root: $ (\sqrt[3]{12x})^3 = 6^3 $ $ 12x = 216 $
3. Isolate $x$ by dividing by 12: $ x = \frac{216}{12} $
4. Calculate the result: $ x = 18 $

The value of $x$ is 18.