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> How would you explain to a 10-year old why 3^0 = 1

You draw the line 3^x. It "passes through" 1 when x = 0. So don't think about the point, think about the line. It's not rigorous but it's intuitive.

http://fooplot.com/index.php?q0=3^x

edit: added link and fixed typos

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This is precisely the problem, though.

The same 10 year old draws two lines: 0^x, and x^0.

They clearly do not meet.

Yeah, I was just talking about the easier concept of x^0. Going the other way, 0^x it makes a lot less sense to me. I know that for all positive x, 0^x = 0. That it pops up to 1 at the origin means the thing is not smooth so drawing lines isn't going to help us.

Plotting z = y^x could be interesting but we're going to see asymptotic behaviour at x = y = 0 (when going in the x direction). So no easy wins there.

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