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Can anyone please tell me how to define $x^\sqrt 2$ ?

If it was $x^2$ or $x ^ \frac{1}{2}$, we could have said that $x^2$ means $x \times x$ and $x ^ \frac{1}{2}$ means a number y such that $y^2 = x$.

But how to define $x^\sqrt 2$ ?

Can anyone please help me ?

Deepak
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anonymous
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    $e^{\sqrt2\ln x}$? – Brian Tung Jun 13 '20 at 05:21
  • Or this: https://math.stackexchange.com/q/1272546/42969 – Martin R Jun 13 '20 at 05:28
  • I think it is poor practice to think of $x^a$ as $x\times x$ $a$ times (or the inverse of this). What does $e^x$ mean in that case? What about $x^i$? Think of the exponential function as a special function that satisfies the property $a^x a^y=a^{x+y}$. Then for integer exponents it follows that the exponential function is repeated multiplication. But you should think of this as a special case of the exponential function and not the other way round. –  Jun 13 '20 at 05:29

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Take a sequence of rationals $q_n$ converging to $\sqrt 2$. Then $x^{\sqrt 2}$ is defined as the limit of the sequence $x^{q_n}$. From continuity of the exponent, this limit exists, is unique, and independent of the choice of sequence. (And as you said, you already have a definition for each term in the sequence, since the powers are rational)

GSofer
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