We can use the identity:
(a + 1/a)^2 = a^2 + 2 + 1/a^2
Squaring both sides of the equation a - 1/a = 7, we get:
(a - 1/a)^2 = 49
Expanding the left-hand side of the equation, we get:
a^2 - 2 + 1/a^2 = 49
Adding 2 to both sides, we get:
a^2 + 1/a^2 = 51
To find the value of a^4 + 1/a^4, we can use the identity:
(a^2 + 1/a^2)^2 - 2 = a^4 + 2 + 1/a^4
Substituting the value of a^2 + 1/a^2 = 51, we get:
(a^2 + 1/a^2)^2 - 2 = 51^2 - 2
Simplifying the right-hand side of the equation, we get:
a^4 + 2 + 1/a^4 = 2600
Therefore, the value of a^4 + 1/a^4 is 2600 - 2, which is equal to 2598.
Hence, a^4 + 1/a^4 = 2598.