The definition of (real) exponent of $b\in\R$, $b>0$ is give in Rudin’s Principles of Mathematical Analysis (3.ed), Chapter 1 Exercise 6 , which says that:

6. Fix $b>1$.

1. If $m,n,p,q$ are integers, $n>0,q>0$, and $r=m/n=p/q$, prove that
   $$(b^m)^{1/n}=(b^p)^{1/q}.$$
   Hence it makes sense to define $b^r=(b^m)^{1/n}$.
2. Prove that $b^{r+s}=b^rb^s$ if $r$ and $s$ are rational.
3. If $x$ is real, define $B(x)$ to be the set of all numbers $b^t$, where $t$ is rational and $t\leq x$. Prove that
   $$b^r=\sup B(r)$$
   when $r$ is rational. Hence it makes sense to define
   $$b^x=\sup B(x)$$
   for every real $x$.
4. Prove that $b^{x+y}=b^xb^y$ for all real $x$ and $y$.

It is a litter hard for beginners to solve this problem, so I will try to post my solution here.

For a. suppose $\alpha=(b^m)^{1/n}$ and $\beta=(b^p)^{1/q}$, then by the definition of $n$th roots, we know that
$$\alpha=(b^m)^{1/n}\Leftrightarrow \alpha^n=b^m, \quad \text{and}\quad \beta=(b^p)^{1/q}\Leftrightarrow \beta^q=b^p.$$
Thus, by the meaning of notation $b^m$ (which says $b$ multiples itself by $m$ times), we know that
$$(b^m)^p=b^{mp}=(b^p)^m,$$
thus
$$\alpha^{np}=(\alpha^n)^p=(b^m)^p=(b^p)^m=(\beta^q)^m=\beta^{qm},$$
since $r=m/n=p/q$, we have $pn=mq$, and
$$\alpha=\beta,$$
by the uniqueness of $n$th roots of a real number. That is $(b^m)^{1/n}=(b^p){1/q}$.

For b. we can assume $r=m/n$ and $s=p/q$ for $n,q>0$ and $m,n,p,q$ are integers. Then by a.,
$$b^{r+s}=b^{\frac{m}{n}+\frac{p}{q}}=b^{\frac{mq+np}{nq}}=(b^{mq+np})^{1/nq}=(b^{mq}b^{np})^{1/nq} =(b^{mq})^{1/nq}(b^{np})^{1/nq}=b^{m/n}b^{p/q}=b^rb^s,$$
we employ the result (prove in the book) that:
$$(\alpha\beta)^{1/n}=\alpha^{1/n}\beta^{1/n},$$
for any $\alpha,\beta\in\R$ and $n$　be a positive integer.

Now, for c., by the definition, we can write $B(x)$ as
$$B(x)=\set{b^t|t\in Q, t\leq x},$$
and note that $b>1$ thus
$$b^t\leq b^r,$$
for all $t,r\in Q$ with $t\leq r$.

In fact, if $b^t>b^r$. Set $t=m/n$, $r=p/q$, and $t\leq r$ then by the definition of ordered field,
$$mq\leq np\Rightarrow b^{mq}\leq b^{np},$$
thus by a.
$$(b^t)^{nq}=b^{mq}\leq b^{np}=(b^{r})^{nq},$$
a contradiction to the assumption $b^t>b^r$.

In conclusion, we know that
$$\sup \set{b^t|t\in Q, t\leq r}\leq b^r,$$
and apparently,
$$b^r\leq\sup B(r)=\sup \set{b^t|t\in Q, t\leq r},$$
thus $b^r=\sup B(r)$.

For d., firstly, for any $r,s\in Q$, if $r\leq x$, $s\leq y$, then by the definition of $b^x, b^y$, we have
$$b^r\leq b^x,\quad b^s\leq b^y,$$
thus
$$b^{r+s}=b^rb^s\leq b^xb^y,$$
since $\R$ is a ordered field.

Now, for any $t\in Q$, if $t\leq x+y$, we can always find $r,s\in Q$ such that $r\leq x, s\leq y$ and $r+s=t$. In fact, by the definition of $x+y$, we know that $x+y=\set{r+s|r\in x, s\in y}$, thus the assertion is certainly satisfied. Consequently,
$$b^{x+y}=\sup\set{b^t|t\in Q, t\leq x+y}\leq b^xb^y.$$
On the other hand, if $b^{x+y}< b^xb^y$, then set $\varepsilon=b^xb^y-b^{x+y} >0$. For any small enough $\delta>0$, by the definition of $b^x, b^y$, there exist $r_0,s_0\in Q$, with $r_0\leq x$, $s_0\leq y$, such that
$$b^{r_0}>b^x-\delta>0,\quad b^{s_0}>b^y-\delta>0.$$
Since $\R$ is an ordered field, then
$$b^{r_0}b^{s_0}>b^xb^y-\delta\left(b^x+b^y-\delta\right),$$
particularly, we can take $\delta$ be the positive roots of
$$\delta\left(b^x+b^y-\delta\right)=\eps/2,$$
then
$$2b^{x+y}\geq 2b^{r_0}b^{s_0} >2b^xb^y-\eps=b^xb^y+b^{x+y},$$
that is
$$b^{x+y}>b^xb^y,$$
contradict to the assumption.

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