Find constant given one specific term

7. (a) Find the first 4 terms, in ascending powers of \(x\), of the binomial expansion of \(( 1 + k x ) ^ { 8 }\), where \(k\) is a non-zero constant. Give each term in its simplest form. Given that the coefficient of \(x ^ { 3 }\) in this expansion is 1512
(b) find the value of \(k\).

3 One of the terms in the binomial expansion of \(( 4 + a x ) ^ { 6 }\) is \(160 x ^ { 3 }\).

1. Find the value of \(a\).
2. Using this value of \(a\), find the first two terms in the expansion of \(( 4 + a x ) ^ { 6 }\) in ascending powers of \(x\).

8. $$g ( x ) = ( 2 + a x ) ^ { 8 } \quad \text { where } a \text { is a constant }$$ Given that one of the terms in the binomial expansion of \(\mathrm { g } ( x )\) is \(3402 x ^ { 5 }\)

1. find the value of \(a\). Using this value of \(a\),
2. find the constant term in the expansion of $$\left( 1 + \frac { 1 } { x ^ { 4 } } \right) ( 2 + a x ) ^ { 8 }$$