Given that:

a^2 - b^2 = 8 ............(1)

ab = 2..................(2)

We need to determine a^4 + b^4

Let us square equation (1).

==> (a^2 - b^2)^2 = 8^2

==> a^4 - 2a^2 b^2 + b^4 = 64

==> a^2 + b^4 = 64 + 2(ab)^2

But from (2) we know that ab= 2

==> a^4 + b^4 = 64 -+2(2^2)

==> a^4 + b^2 = 64 + 8 = 72

**==> a^4 + b^4 = 72**

We have a^2 - b^2 = 8 and ab = 2

Now (a - b)^2 = a^2 + b^2 + 2ab

=> (a^2 - b^2)^2 = a^4 + b^4 - 2*a^2*b^2

a^4 + b^4

=> (a^2 - b^2)^2 + 2*a^2*b^2

=> 8^2 + 2* (ab)^2

=> 64 + 2*2^2

=> 64 + 8

=> 72

**Therefore a^4 + b^4 = 72**

Q:If a^2 - b^2 = 8 and a*b = 2, find a^4 + b^4 (edited ).

A:

a^2-b^2 = 8

Therefore (a^2+b^2)^2 = 8^2 = 64.

=> a^4+b^4-2a^2b^2 = 64.

=> a^4+b^4 = 64 + 2a^2b^2

= > a^4+b^4 = 64+ 2(ab)^2

=> a^4+b^4= 64 +2(2)^2, as ab = 2 by data.

=> a^4+b^4 = 64+2*4 = 64 + 8 = 72.

=> a^4+b^4 = 72.

**Therefore a^4+b^4 = 72.**