Let u and v be two strings. What about the reverse order of their concatenaited string?
No, $(u\cdot v)^R\not=u^R\cdot v^R$ more often than not. 

For example, if $u$ is the string $race$ and $v$ is the string $car$. Then  $$(u\cdot v)^R=(racecar)^R=racecar$$
while
$$u^R\cdot v^R=(race)^R(car)^R=ecarrac.$$ 



Here are a few related exercises. All variables stand for strings.

Exercise 1. If $u$ or $v$ is the empty word, then $(u\cdot v)^R=u^R\cdot v^R$.

Exercise 2. If $u$ or $v$ are words of length 1 such that $(u\cdot v)^R=u^R\cdot v^R$. Then $u=v$.

Exercise 3. If $u=u^R$ and $v=v^R$, can we guarantee $(u\cdot v)^R=u^R\cdot v^R$?

Exercise 4. Prove that $(u\cdot v)^R=v^R\cdot u^R$.