这个.应该是
There is not an exact definition of the word happiness.
要不就是
There is no exact definition of the word happiness.
an 是因为后面的单词是 exact,开始音是元音.
the 是特指一个词.
ps 楼主,第二个一定要用the 因为后面跟了 happiness,这里的word 特指happiness这个词……