The operation ⊕ between two positive integers a and b is defined by the rule: a ⊕ b = sqrt(a) + sqrt(b). If 324 ⊕ 289 = 35 and 441 ⊕ 484 = 43, then what is the value of 625 ⊕ 400?

Introduction / Context:
This is a definition-based problem involving a custom operation denoted by the symbol ⊕. The definition of the operation is given explicitly in terms of square roots, and several examples are provided. The task is to apply the same rule to a new pair of numbers to find the correct result. This type of question assesses how well you understand and use function definitions.

Given Data / Assumptions:

• The operation is defined as a ⊕ b = sqrt(a) + sqrt(b) for positive integers a and b.
• We are told that 324 ⊕ 289 = 35 and 441 ⊕ 484 = 43 as illustrations of the rule.
• We must compute 625 ⊕ 400 using the same definition.

Concept / Approach:
We treat ⊕ as a shorthand for adding the square roots of the two numbers. So to find 625 ⊕ 400, we simply compute sqrt(625) and sqrt(400) and add them together. Because the numbers given are perfect squares, their square roots are integers, which makes the calculation straightforward.

Step-by-Step Solution:
From the definition, a ⊕ b = sqrt(a) + sqrt(b).For the example 324 ⊕ 289, we have sqrt(324) = 18 and sqrt(289) = 17, so 18 + 17 = 35, which matches the given value 35.For 441 ⊕ 484, sqrt(441) = 21 and sqrt(484) = 22, so 21 + 22 = 43, again matching the example.Now apply the same rule to 625 ⊕ 400.Compute sqrt(625) = 25 and sqrt(400) = 20.Add the results: 25 + 20 = 45.

Verification / Alternative check:
We can quickly verify that 625 and 400 are indeed perfect squares: 25 × 25 = 625 and 20 × 20 = 400.Therefore, there is no ambiguity in the square roots, and the sum 45 is exact, not approximate.

Why Other Options Are Wrong:
40 would correspond to a situation where one of the square roots was incorrectly taken as 15 instead of 20 or 25, which does not fit the definitions of square roots.35 is the result of 18 + 17 from the first example and does not relate to the pair 625 and 400.30 is smaller than either of the individual square roots 25 and 20, so it cannot be their sum.

Common Pitfalls:
One common mistake is to add the numbers themselves (for example, 625 + 400 = 1025) and then take a square root, which reverses the defined order of operations.Another pitfall is to miscalculate a square root, especially if you forget that 625 and 400 are standard perfect squares and try to approximate them incorrectly.

Final Answer:
According to the definition a ⊕ b = sqrt(a) + sqrt(b), we have 625 ⊕ 400 = 25 + 20 = 45.