Question

If \(a b c d=81\), what is the minimum value of \(a+b+c+d ?\) (a) 18 (b) 9 (c) 12 (d) 36

Short answer

Short Answer: The minimum value of the sum of the four variables a, b, c, and d equals 12, a result we obtained by applying the AM-GM inequality. This value occurs when all four numbers are equal (a = b = c = d = 3).

Step-by-step solution

1. Apply AM-GM inequality to the given numbers

Since we want to minimize the sum \(a+b+c+d\), we'll use AM-GM inequality to relate the sum to their product. For the four non-negative numbers \(a, b, c, d\), the AM-GM inequality is given by: \(\frac{a+b+c+d}{4} \geq \sqrt[4]{abcd}\) Here, we have the product of the four numbers, \(abcd=81\). We want to find the minimum value of their sum, \(a+b+c+d\).

2. Replace the product of the variables with the given value

Now we replace the product \(abcd\) with the given value, 81, in the inequality: \(\frac{a+b+c+d}{4} \geq \sqrt[4]{81}\)

3. Calculate the geometric mean

Calculate the fourth root of 81: \(\sqrt[4]{81} = 3\)

4. Rewrite the inequality in terms of the sum

The inequality now becomes: \(\frac{a+b+c+d}{4} \geq 3\)

5. Solve for the sum

Now we solve the inequality for the sum \(a+b+c+d\): \(a+b+c+d \geq 12\) Therefore, the minimum value of the sum \(a+b+c+d\) is \(\boxed{12}\) (option (c)). Note that the equality holds when all four numbers are equal, i.e., \(a=b=c=d=3\).

Key concepts

Geometric Mean

The geometric mean often comes into play when dealing with sets of numbers, especially in context of the AM-GM inequality. The term "geometric mean" refers to the symmetrical average of a set of numbers. In simpler terms, it's like finding a middle value that represents all numbers, but in a multiplicative way.

For example, if you have four numbers, say \(a, b, c,\) and \(d\), the geometric mean is calculated by multiplying all the numbers together and then taking the fourth root. Mathematically, it's expressed as:

• Geometric Mean = \(\sqrt[4]{abcd}\)

This implies that when you have a number like 81 in this context, you are essentially asking 'what number, if multiplied by itself four times, gives 81?'

Understanding the geometric mean is key when finding the minimum value using the AM-GM inequality. It helps bridge the gap between the sum and product of numbers, allowing us to relate and manipulate them efficiently.

Fourth Root

The concept of a fourth root can be a tricky idea at first, but it is simpler when you visualize it as reversing an exponentiation. Similar to how a square root undoes squaring, the fourth root undoes raising to the fourth power.

So, when you have a number like 81 and you want to find its fourth root, you're looking for a number which, when multiplied by itself four times, results in 81.

• Fourth Root of 81 = 3, because \(3^4 = 3 \times 3 \times 3 \times 3 = 81\)

By calculating the fourth root of the product of numbers \(abcd\), we simplify the AM-GM inequality to work out the possible minimum for their sum. It provides a crucial step to reach this minimum, effectively relating multiplication to addition in an understandable way.

Minimum Value

Finding the minimum value in mathematics involves using inequalities and properties that constrain outputs. In this particular exercise, we use the AM-GM inequality to establish limits on the possible sum of our variables.

The AM-GM (Arithmetic Mean-Geometric Mean) inequality states that the arithmetic mean of non-negative numbers is always greater than or equal to the geometric mean. Essentially, it provides a roadmap to finding the most optimized result in balancing a sum with known product boundaries.

• Formula: \( \frac{a+b+c+d}{4} \geq \sqrt[4]{abcd} \)
• Replace \( abcd = 81 \), we get \( \frac{a+b+c+d}{4} \geq 3 \)

Rearranging gives \(a+b+c+d \geq 12\), indicating that 12 is the minimum sum if all elements are equal to the fourth root. Hence, the answer is 12 when each variable equals 3. This result emphasizes how the AM-GM inequality optimizes numerical arrangements to achieve minimal sum configurations.