LHS = (a + 1)7 (b + 1)7 (c + 1)7
= [(a + 1) (b + 1) (c + 1)]7
= [1 + a + b + c + ab + bc + ca + abc]7 > (a + b + c + ab + bc + ca + abc)7 …(i)
Now using the AM, GM inequality, i.e., AM > GM, we have
\(\frac{a+b+c+ab+bc+ca+abc}{7}\) > (a.b.c.ab.bc.ca.abc)\(\frac17\)
⇒ \(\frac{1}{7^7}\)(a + b + c + ab + bc + ca + abc)7 (a4 b4 c4) [Raising both sides to power 7]
⇒ (a + b + c + ab + bc + ca + abc)7 > 77 (a4 b4 c4) …(ii)
From (i) and (ii)
(a + 1)7 (b + 1)7 (c + 1)7 > 77 (a4 b4 c4).