**Product of (a-1/a)(a 1/a)(a raised2 1/a raised2)(a raised4 1/a raised4) using suitable identities**

To find the product of the given expression, we can simplify each term using suitable algebraic identities and then multiply them together. Let's break down the expression and simplify each term step by step:

1. (a - 1/a):
- We can simplify this term by multiplying both the numerator and denominator by 'a', which gives us:
(a^2 - 1) / a

2. (a + 1/a):
- Similarly, we can multiply both the numerator and denominator by 'a', which gives us:
(a^2 + 1) / a

3. (a raised2 + 1/a raised2):
- This term can be simplified using the identity (a + b)(a - b) = a^2 - b^2. In this case, let's consider 'a' as 'a' and 'b' as '1/a':
(a + 1/a)(a - 1/a) = a^2 - (1/a)^2
= a^2 - 1/a^2

4. (a raised4 + 1/a raised4):
- We can use the same identity as above, considering 'a' as 'a^2' and 'b' as '1/a^2':
(a^2 + 1/a^2)(a^2 - 1/a^2) = a^4 - (1/a^4)
= a^4 - 1/a^4

Now, let's multiply all the simplified terms together:

((a^2 - 1) / a) * ((a^2 + 1) / a) * (a^2 - 1/a^2) * (a^4 - 1/a^4)

Simplifying this expression further, we can combine the numerators and denominators:

((a^2 - 1)(a^2 + 1)(a^2 - 1/a^2)(a^4 - 1/a^4)) / (a^2 * a * a^2 * a^4)

Next, let's expand the numerator:

(a^4 - 1)(a^4 - 1/a^4)

Using the identity (a^2 - b^2)(a^2 + b^2) = a^4 - b^4, we can simplify the numerator further:

(a^4 - 1)(a^4 - 1/a^4) = (a^4)^2 - (1/a^4)^2
= a^8 - 1

Finally, let's substitute this simplified expression back into the main expression:

(a^8 - 1) / (a^2 * a * a^2 * a^4)

Now, we can combine the exponents and simplify further:

(a^8 - 1) / (a^8 * a^4)
= 1/a^8 - 1/a^12

Therefore, the product of (a-1/a)(a + 1/a)(a^2 + 1/a^2)(a^4 + 1/a^4) is 1/a^