If `abcd=81`, what is the minimum value of `a+b+c+d`?

Options

A. 18
B. 9
C. 12
D. 36

Answer

To solve the problem where \( abcd = 81 \) and we need to find the minimum value of \( a + b + c + d \), we can use the concept of the Arithmetic Mean-Geometric Mean (AM-GM) inequality.

### Step-by-Step Solution:

1. **Understand the Given Information**: We know that the product of four numbers \( a, b, c, d \) is equal to 81, i.e., \( abcd = 81 \).

2. **Apply the AM-GM Inequality**: The AM-GM inequality states that for any non-negative real numbers \( x_1, x_2, \ldots, x_n \):
\[ \frac{x_1 + x_2 + \ldots + x_n}{n} \geq \sqrt[n]{x_1 x_2 \ldots x_n} \]
For our case with four variables \( a, b, c, d \):
\[ \frac{a + b + c + d}{4} \geq \sqrt[4]{abcd} \]

3. **Substitute the Known Product**: Since \( abcd = 81 \), we can substitute this into the inequality:
\[ \frac{a + b + c + d}{4} \geq \sqrt[4]{81} \]

4. **Calculate the Fourth Root**: We know that \( 81 = 3^4 \), so:
\[ \sqrt[4]{81} = \sqrt[4]{3^4} = 3 \]

5. **Rewrite the Inequality**: Now we can rewrite our inequality:
\[ \frac{a + b + c + d}{4} \geq 3 \]

6. **Multiply Both Sides by 4**: To isolate \( a + b + c + d \), multiply both sides by 4:
\[ a + b + c + d \geq 12 \]

7. **Conclusion**: The minimum value of \( a + b + c + d \) is 12. This minimum occurs when \( a = b = c = d \). Since \( abcd = 81 \), if \( a = b = c = d = 3 \), then \( 3 \times 3 \times 3 \times 3 = 81 \).

### Final Answer:
The minimum value of \( a + b + c + d \) is **12**.