What is the geometric mean of 10, 40 and 60?

Options

A. `10root(3)(3)`
B. `20root(3)(3)`
C. `40root(3)(3)`
D. `70root(3)(3)`

Answer

To find the geometric mean of the numbers 10, 40, and 60, we follow these steps:

### Step-by-Step Solution:

1. **Identify the Numbers**: We have three numbers: A = 10, B = 40, and C = 60.

2. **Formula for Geometric Mean**: The geometric mean (GM) of n numbers is given by the formula:
\[ GM = (A \times B \times C)^{\frac{1}{n}} \]
where n is the number of values. In this case, n = 3.

3. **Substitute the Values**: Substitute the values of A, B, and C into the formula:
\[ GM = (10 \times 40 \times 60)^{\frac{1}{3}} \]

4. **Calculate the Product**: First, calculate the product of the numbers:
\[ 10 \times 40 = 400 \]
\[ 400 \times 60 = 24000 \]
So, the product is 24000.

5. **Take the Cube Root**: Now, we need to find the cube root of 24000:
\[ GM = (24000)^{\frac{1}{3}} \]

6. **Simplify the Expression**: We can factor 24000 to make it easier to find the cube root:
\[ 24000 = 24 \times 1000 = 24 \times 10^3 \]
We know that \( 24 = 2^3 \times 3 \), so:
\[ 24000 = 2^3 \times 3 \times 10^3 \]

7. **Apply the Cube Root**: Now apply the cube root:
\[ GM = (2^3 \times 3 \times 10^3)^{\frac{1}{3}} = 2 \times 3^{\frac{1}{3}} \times 10 \]

8. **Calculate the Final Value**:
\[ GM = 20 \times 3^{\frac{1}{3}} \]

### Final Answer:
Thus, the geometric mean of 10, 40, and 60 is:
\[ GM = 20 \times 3^{\frac{1}{3}} \]