Given the languages, let us see what strings are generated
$L_1= a^ib^ic$ ($i >=0$)
It generates below strings
$\left \{c, abc, aabbc, aaabbbccc, aaaabbbbc \right \}$
Above are strings which begin with $a$'s followed by equal number of $b$'s and end with single $c$
$L_2= ab^ic^i$ ($i >=0$)
It generates below strings
$\left \{a, abc, abbcc, abbbccc, abbbbcccc \right \}$
Above are strings which begin with $a$ followed by $b$'s followed by equal number of $c$'s
So, the intersection of above languages would contain
only the string $\left \{abc \right \}$ as no other string is common between these languages
So, the language would be
$L= abc$