10. (a) A student's attempt to answer the question
"Prove by contradiction that if \(n ^ { 3 }\) is even, then \(n\) is even" is shown below. Line 5 of the proof is missing. Assume that there exists a number \(n\) such that \(n ^ { 3 }\) is even, but \(n\) is odd. If \(n\) is odd then \(n = 2 p + 1\) where \(p \in \mathbb { Z }\)
So \(n ^ { 3 } = ( 2 p + 1 ) ^ { 3 }\) $$\begin{aligned} & = 8 p ^ { 3 } + 12 p ^ { 2 } + 6 p + 1
& = \end{aligned}$$ This contradicts our initial assumption, so if \(n ^ { 3 }\) is even, then \(n\) is even. Complete this proof by filling in line 5.
(b) Hence, prove by contradiction that \(\sqrt [ 3 ] { 2 }\) is irrational.