Okay, so I need to figure out the value of 7/30 @ 10/21 using the given operation. Let me start by understanding how the @ operation works. The definition says that for simplified fractions m/n and p/q, the operation is (m)(p)(q/n). Hmm, let me parse that again. So, if we have two fractions, the first one is m/n and the second one is p/q. Then the result is m multiplied by p multiplied by (q divided by n). But wait, q/n is a fraction, so that would be m * p * (q/n). Let me write that down to make it clearer.

So, the operation @ takes two fractions and combines them as follows: numerator1 * numerator2 * (denominator2 / denominator1). But wait, the denominator2 divided by denominator1. Let me confirm that. The definition is (m)(p)(q/n). So m is the numerator of the first fraction, p is the numerator of the second fraction, q is the numerator of the second fraction's simplified form, and n is the denominator of the first fraction. Wait, no, hold on. Let me check the original definition again.

The operation is defined as (m/n) @ (p/q) = m * p * (q/n). So, yes, m is the numerator of the first fraction, p is the numerator of the second fraction, q is the numerator of the second fraction, and n is the denominator of the first fraction. So, when we perform the operation, we take the numerators of both fractions, multiply them together, and then multiply by the denominator of the second fraction divided by the denominator of the first fraction. So essentially, it's (m * p * q) / n. But wait, let me make sure.

Wait, m * p * (q/n) is the same as (m * p * q) / n. So that's the same as (m * p * q) divided by n. So the result is a single fraction with numerator m*p*q and denominator n. But the problem says the fractions are simplified. So, the fractions m/n and p/q are simplified, meaning that m and n are coprime, and p and q are coprime. But does that affect the operation? Let me see.

Wait, the fractions given are 7/30 and 10/21. Let me check if they are simplified. 7 and 30 have no common factors except 1, so 7/30 is simplified. 10 and 21 also have no common factors except 1, so 10/21 is simplified. So, m = 7, n = 30, p = 10, q = 21. So plugging into the formula: m * p * (q/n) = 7 * 10 * (21/30). Let me compute that step by step.

First, calculate 7 * 10. That's 70. Then multiply by 21/30. So 70 * (21/30). Let me simplify 21/30 first. 21 and 30 have a common factor of 3. Dividing numerator and denominator by 3 gives 7/10. So 70 * (7/10) = (70 * 7) / 10. 70 divided by 10 is 7, so 7 * 7 = 49. Therefore, the result is 49. But wait, the problem says "simplified value," so 49 is an integer, which is already simplified. So the answer is 49. Hmm, that seems straightforward, but let me double-check to make sure I didn't make a mistake.

Alternatively, let me compute 7/30 @ 10/21 step by step. According to the definition, it's (7)(10)(21/30). Let's compute 21/30 first. 21 divided by 30 is 0.7, but as a fraction, it's 7/10. Then multiply 7 * 10 * 7/10. The 10 cancels out with the 10 in the numerator, leaving 7 * 7 = 49. So that's correct. Alternatively, if I do the multiplication without simplifying first: 7*10=70, 70*21=1470, then divide by 30: 1470/30. Let's divide numerator and denominator by 10: 147/3. Then divide 147 by 3: 49. Same result. So that's correct.

Wait, but the problem says "simplified fractions p/q" in the definition. So when we apply the operation, do we need to ensure that the second fraction is simplified? Wait, in the problem statement, the fractions given are 7/30 and 10/21, which are already simplified. So p/q is already in simplest terms. So in the formula, q is the numerator of the simplified second fraction, which is 21, and n is the denominator of the first fraction, which is 30. So that's correct.

But let me just confirm once more. The operation is defined for simplified fractions, so when we use the operation, we have already simplified the fractions. Therefore, when substituting into the formula, m/n and p/q are already in simplest terms. Therefore, in this case, m=7, n=30, p=10, q=21. So the calculation is 7*10*(21/30) = 70*(7/10) = 49. Yes, that's correct.

Alternatively, another way to look at it: 7/30 @ 10/21 = (7*10) / (30/21). Wait, is that another way to represent it? Let me check. If we take (m)(p) = 70, and then divide by (n/q) = 30/21. So 70 divided by (30/21) is 70 * (21/30) = same as before, 49. So either way, the result is 49.

Therefore, the simplified value is 49. Since 49 is an integer, it's already in its simplest form. So the answer is 49.

Wait, just to make sure there's no trick here. The problem defines @ for simplified fractions, and the fractions given are simplified. So the operation is applied correctly. The calculation steps are correct. The answer is 49. I think that's it.
Therefore, the simplified value of $\frac{7}{30}@\frac{10}{21}$ is $\boxed{49}$.
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