Bra-ket notation, Bits, & Superposition I am a quantum computing enthusiast, and recently I stumbled upon this the following two propositions:
$$ \alpha|1\rangle + \beta|0\rangle$$
What does this mean? 
My understanding of this is that: the two bits, 1 and 0 are represented in a state of superposition, hence the bra-ket notation (which is commonly used for quantum mechanics), i.e., this is a qubit.
Or is there a more concise explanation of this?
Also:
$$(\alpha|1\rangle + \beta|0\rangle)^N$$
What does it mean to raise this quantity (of superimposed bits, i.e., qubit) to the $N$th degree? If we take $2^N$ where $N$ is the number of qubits then this tells us the number of bits in the desired number of qubits.
Is what I have stated in this post, generally correct?
 A: The expression
\begin{align}
  \alpha|1\rangle+\beta|0\rangle
\end{align}
is the state of a single qubit written as a linear combination of the state $|1\rangle$ and the state $|0\rangle$.  If you were to make a measurement on this qubit, then you would either return $1$ or $0$ with probabilities $|\alpha|^2$ and $|\beta|^2$ respectively.
The expression
\begin{align}
  (\alpha|1\rangle + \beta|0\rangle)^N
\end{align}
is probably a shorthand for
\begin{align}
  \underbrace{(\alpha|1\rangle + \beta|1\rangle)\otimes\cdots \otimes(\alpha|1\rangle + \beta|0\rangle)}_{N\,\text{factors}},{}{}
\end{align}
namely the $N$-fold tensor product of the state $\alpha|1\rangle + \beta|1\rangle$ with itself.  This represents the state of $N$ qubits.
If you make a measurement on a system of $N$ such qubits, then you will obtain one of $2^N$ possibilities, namely the $2^N$ distinct sequences of $1$'s and $0$'s obtained by expanding out the product.  The probability of obtaining such a sequence is its associated coefficient.  In fact, in this case, the probability of measuring a single such sequence is $|\alpha|^n|\beta|^{N-n}$ where $n$ is the number of $1$'s in the sequence, and therefore $N-n$ is the number of $0$'s in the sequence.  So, for example, the state
\begin{align}
  |1\rangle|1\rangle\underbrace{|0\rangle\cdots |0\rangle}_{N-2\,\text{factors}}
\end{align}
has associated probability $|\alpha|^2|\beta|^{N-2}$ of measurement.
