Solve for $a$: ​

Note that  

√[ 4 +  √ [16 + 16a] ]     can be written as

√[ 4 +  4√ [1 + a] ]   =

2 √[ 1+  √ [1 + a] ]

Let this  = 2x

So we have

2x  + x  =  6

3x  = 6

x = 2

So...

√[ 1+  √ [1 + a] ]    = 2     square both sides

1 + √ [1 + a]  =  4          subtract 1 from both sides

√ [1 + a]   = 3          square both sides again

1 + a  =  9    →    a  =  8