Okay, let's see. The problem is to find the value of 513 squared minus 487 squared. Hmm, I remember there's a formula for the difference of squares. Let me recall... Oh right, the difference of squares formula is a² - b² = (a - b)(a + b). So maybe I can apply that here instead of calculating each square separately. That might save some time and reduce the chance of errors.

First, let me identify a and b in this problem. Here, a is 513 and b is 487. So applying the formula, it becomes (513 - 487)(513 + 487). Let me compute those two parts step by step.

Starting with the subtraction: 513 minus 487. Let me do that. 513 minus 400 is 113, then minus 87 more. So 113 minus 87. 113 minus 80 is 33, minus 7 is 26. So 513 - 487 equals 26. Okay, that seems right.

Now the addition part: 513 plus 487. Let me add those. 500 plus 400 is 900, 13 plus 87 is 100. So 900 plus 100 is 1000. Wait, that's not right. Wait, 513 + 487. Let's break it down. 500 + 400 = 900, 13 + 87 = 100. So total is 900 + 100 = 1000. Oh, actually, that is correct. Because 513 + 487 = 1000. That's a nice round number. So the two factors are 26 and 1000.

So multiplying those together: 26 * 1000. That's straightforward. 26 times 1000 is 26,000. Therefore, the value of 513² minus 487² is 26,000. 

Wait, let me verify this to make sure I didn't make any mistakes. Let me compute 513 squared and 487 squared separately and subtract them to check.

First, 513 squared. Let's compute that. 513 * 513. Hmm, that's a bit tedious. Let me use the formula (a + b)² = a² + 2ab + b². Let's split 513 into 500 + 13. So (500 + 13)² = 500² + 2*500*13 + 13². Calculating each term:

500² = 250,000

2*500*13 = 1000*13 = 13,000

13² = 169

Adding them up: 250,000 + 13,000 = 263,000; 263,000 + 169 = 263,169. So 513² is 263,169.

Now 487 squared. Let's do the same. Split 487 into 400 + 87. (400 + 87)² = 400² + 2*400*87 + 87².

400² = 160,000

2*400*87 = 800*87. Let's compute 800*80 = 64,000 and 800*7=5,600. So 64,000 + 5,600 = 69,600.

87². Let's compute that. 80² = 6,400, 7²=49, and 2*80*7=1,120. So 6,400 + 1,120 = 7,520; 7,520 + 49 = 7,569.

Now add them all up: 160,000 + 69,600 = 229,600; 229,600 + 7,569 = 237,169. So 487² is 237,169.

Subtracting the two: 263,169 - 237,169. Let's do that. 263,169 minus 237,169. The 169 cancels out, leaving 263,000 - 237,000 = 26,000. So that matches the previous result. Therefore, the answer is indeed 26,000.

Alternatively, maybe there's another way to check. Since 513 and 487 are 26 apart and their average is (513 + 487)/2 = 1000/2 = 500. So the difference of squares is (26)(1000) = 26,000. Yep, same answer.

So both methods confirm that the answer is 26,000. Therefore, I can be confident that the correct answer is 26,000.
Thus, the value of $513^2 - 487^2$ is $\boxed{26000}$.
</think>