If a, b, c are positive real numbers such that a2b3c4 = 21933, the minimum value of a + b + c is _______.

This problem can be solved using the AM-GM inequality (Arithmetic Mean-Geometric Mean Inequality). The AM-GM inequality states that the arithmetic mean of non-negative real numbers is greater than or equal to the geometric mean.

Step 1: Rewrite the equation a^2b^3c^4 = 21933 as aabbbcccc = 21933.

Step 2: Apply the AM-GM inequality. We have 10 variables (2 a's, 3 b's, and 4 c's), so the arithmetic mean is (2a + 3b + 4c)/10.

Step 3: The geometric mean is the 10th root of (aabbbcccc), which is the 10th root of 21933.

Step 4: According to the AM-GM inequality, (2a + 3b + 4c)/10 >= 10th root of 21933.

Step 5: Multiply both sides by 10 to get 2a + 3b + 4c >= 10 * (10th root of 21933).

Step 6: The minimum value of a + b + c is achieved when 2a = 3b = 4c. So, we can write a = 3b/2 and c = 3b/4.

Step 7: Substitute these values into the inequality to get 2*(3b/2) + 3b + 4*(3b/4) >= 10 * (10th root of 21933).

Step 8: Simplify to get b >= 10 * (10th root of 21933).

Step 9: Since a = 3b/2 and c = 3b/4, we can substitute these values into the equation a + b + c to get the minimum value of a + b + c.

Step 10: The minimum value of a + b + c is therefore 10 * (10th root of 21933).