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We have ${a}^{2} - {b}^{2} = 8$, which we can square both sides to get $a^4 + b^4 - 2a^2b^2 = 8^2$. Since $ab = 2$, we can square both sides again to get $a^2b^2 = 2^2$. Then, we can subsitute that value into the equation; $a^4 + b^4 - 2(4) = 8^2$ -> $a^4 + b^4 - 8 = 64$ -> $a^4 + b^4 = 72$.

- Daisy

a^2  - b^2   = 8   and  ab  = 2 

So 

(a^2 - b^2) (a^2 - b^2)  =  (8) * (8)

a^4  - 2(ab)^2 + b^4  = 64

a^4 + b^4  = 64 + 2(ab)^2

a^4 + b^4  =  64 + 2(2)^2

a^4 + b^4  = 64 + 8

a^4 + b^4  = 72