Okay, let's see. I need to find the distance between two points, right? The points are (2, -6) and (-4, 3). Hmm, I remember there's a formula for the distance between two points in a coordinate plane. Let me try to recall. Oh yeah, it's something like the square root of the sum of the squares of the differences in each coordinate. So, if the points are (x1, y1) and (x2, y2), the distance is sqrt[(x2 - x1)^2 + (y2 - y1)^2]. Let me make sure that's correct.

Wait, so first I need to subtract the x-coordinates and the y-coordinates of the two points, square those differences, add them together, and then take the square root. That sounds right. Let me write that down step by step to avoid mistakes.

First, identify the coordinates. The first point is (2, -6), so x1 is 2 and y1 is -6. The second point is (-4, 3), so x2 is -4 and y2 is 3. Now, plug these into the distance formula.

Calculating the differences: x2 - x1 is -4 - 2, which is -6. Similarly, y2 - y1 is 3 - (-6), which is 3 + 6 = 9. Wait, that's right. So the differences are -6 and 9. Now square those. (-6)^2 is 36, and 9^2 is 81. Adding those together gives 36 + 81 = 117. Then the distance is the square root of 117.

But the problem says to express the answer in simplest radical form. So I need to simplify sqrt(117). Let me factor 117 to see if there are any square factors. 117 divided by 9 is 13, so 117 = 9 * 13. Then sqrt(117) = sqrt(9*13) = sqrt(9) * sqrt(13) = 3*sqrt(13). That should be the simplified form.

Wait, let me double-check each step to make sure I didn't make any errors. First, the differences: x2 - x1 is indeed -4 - 2 = -6. Then y2 - y1 is 3 - (-6) = 3 + 6 = 9. Squaring those gives 36 and 81. Adding them gives 117. Square root of 117 factors into 9*13, so sqrt(9*13) is 3*sqrt(13). Yeah, that seems correct.

Alternatively, I could check using the distance formula again. Let me verify with another method. Suppose I plot these points on a coordinate plane. The horizontal distance between them is from x=2 to x=-4, which is a distance of | -4 - 2 | = 6 units. The vertical distance is from y=-6 to y=3, which is |3 - (-6)| = 9 units. Then, the distance between the points is the hypotenuse of a right triangle with legs 6 and 9. The hypotenuse would be sqrt(6^2 + 9^2) = sqrt(36 + 81) = sqrt(117) = 3*sqrt(13). Yep, same result.

So I think that's correct. The distance is 3*sqrt(13) units. Let me just confirm once more that I didn't mix up any numbers. The original points: (2, -6) and (-4, 3). Differences: x: from 2 to -4 is a change of -6, y: from -6 to 3 is a change of +9. Squares: 36 and 81. Sum: 117. Square root: 3*sqrt(13). All steps check out. I don't see any mistakes here. So the answer should be 3√13.
Therefore, the distance between the points $(2, -6)$ and $(-4, 3)$ is $\boxed{3\sqrt{13}}$ units.
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