How do you raise a number to a power that is an irrational number? Such as 2<sup>pi</sup>. I understand how to raise something to a power that's a fraction, but you can't really make pi into a fraction, so I don't think it can be done that way.

How do you raise a number to a power that is an irrational number? Such as 2<sup>pi</sup>. I understand how to raise something to a power that's a fraction, but you can't really make pi into a fraction, so I don't think it can be done that way.
You use "continuity" and the fact that the rationals are "dense" in the reals. Instead of defining each of these words, let me put it this way: if you fix a number a and want to raise it to the power x, you find a sequence of rational numbers x<sub>n</sub> that converges to x. Then the sequence a<sup>x<sub>n</sub></sup> converges, and we define a<sup>x</sup> to be the limit of this sequence. You can also think of this as "filling in the holes" of the graph you get when you evaluate y = a<sup>x</sup> for rational values of x.
I can go on if you wish.
ok, I think I have it. If you don't mind, could you give a simple example so I can make sure I have it? Only if you feel like it, I don't want to be a pain.
So if I wanted to calculate 2<sup>π</sup>, I find a sequence of rationals converging to pie by cutting off its decimal expansion. Then I evaluate 2 to each of these powers. I take the limit, and I get:
2<sup>3</sup> = 8
2<sup>3.1</sup> = 8.5741877...
2<sup>3.14</sup> = 8.81524093...
2<sup>3.141</sup> = 8.8213533...
2<sup>3.1415</sup> = 8.82441108...
2<sup>3.14159</sup> = 8.8249616...
2<sup>3.141592</sup> = 8.82497383...
2<sup>3.1415926</sup> = 8.8249775...
2<sup>3.14159265</sup> = 8.82497781...
2<sup>3.141592654</sup> = 8.82497783...
...
2<sup>π</sup> = 8.82497783...
This is valid for a mathematical explanation of what 2<sup>π</sup> is. But we don't actually calculate exponents like this. We (probably) calculate (or, rather, approximate) them using the power series for e<sup>x</sup> and then using that 2<sup>x</sup> = e<sup>x ln 2</sup> (and using power series for ln).
So the idea of cutting off the decimal expansion is to make the power you're raising to rational, so you can calculate, for example, <sup>50</sup>rt(2<sup>157</sup>) for 2<sup>3.14</sup>? And then what's the setup for the limit exactly? I won't even attempt the way you said it's really done, I think I have too much lack of background knowledge to tackle that right now.
Yeah, cutting off a decimal expansion gives a rational approximation. There are "better" approximations, but these are the easiest.
The limit is more of a mathematical formality. It gives meaning to the expression a<sup>b</sup>, and it tells you how you can approximate this expression, but you can't actually sit down and calculate it. You can approximate a<sup>b</sup> by a<sup>c</sup> if c is rational and really close to b. But then you need some way of calculating a<sup>c</sup>. As I suggested before, this is not as easy as it looks.
Got it. Thanks.
If a>0, . That's what I take to be the definition of for any real number x.
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