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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

$L= abc$
edited by
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plz make me correct if i m wrong.........

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