為何x^(p/q)會被定義為q√(x^p)?

First, I assume you know the law of rational indices.
i.e. x^(p+q) = x^p * x^q .... and so on.

Here I will give the non-rigorous proof of the theorem x^(p/q)=q√(x^p)

First, we define the number x^(1/q),
We use the fact that (I don't want to prove here) for any positive real number x, there exists an unique positive real number y such that y^q = x, the solution
y = x^(1/q)= q√x is therefore defined as the solution of that equation.

Next, we can treat x^p to be one real number, we can claim that for any positive real number x, there exists an unique positive real number y such that y^q = x^p, the solution
y = (x^p)^(1/q) = q√x^p is therefore defined as the solution of that equation.

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For the second question, we can consider the case r to be rational first
When r is an positive integer, for example, we could interpret 2^3 = 2 * 2 * 2
(i.e. 2 multiply itself 3 times) = 8

When r is rational (i.e. r = p/q), we could use the fact x^(1/q) = q√x
Hence we could interpret 2^(3/2) = 2^3 * 2*(1/2) = 2 * 2 * 2 * (2√2) = 4 * √2

However, if q is irrational (say, for example 2^(√2)), we should interpret it as a limit.
More precisely, it is the limit of the sequence { 2^1, 2^(1.4), 2^(1.41), 2^(1.414) ...}
So the meaning of x^q can be explained in similar manner, in case q is irrational.

First, I assume you know the law of rational indices.
i.e. x^(p+q) = x^p * x^q .... and so on.

Here I will give the non-rigorous proof of the theorem x^(p/q)=q√(x^p)

First, we define the number x^(1/q),
We use the fact that (I don't want to prove here) for any positive real number x, there exists an unique positive real number y such that y^q = x, the solution
y = x^(1/q)= q√x is therefore defined as the solution of that equation.

Next, we can treat x^p to be one real number, we can claim that for any positive real number x, there exists an unique positive real number y such that y^q = x^p, the solution
y = (x^p)^(1/q) = q√x^p is therefore defined as the solution of that equation.

For the second question, we can consider the case r to be rational first
When r is an positive integer, for example, we could interpret 2^3 = 2 * 2 * 2
(i.e. 2 multiply itself 3 times) = 8

When r is rational (i.e. r = p/q), we could use the fact x^(1/q) = q√x
Hence we could interpret 2^(3/2) = 2^3 * 2*(1/2) = 2 * 2 * 2 * (2√2) = 4 * √2

However, if q is irrational (say, for example 2^(√2)), we should interpret it as a limit.
More precisely, it is the limit of the sequence { 2^1, 2^(1.4), 2^(1.41), 2^(1.414) ...}
So the meaning of x^q can be explained in similar manner, in case q is irrational.