Final answer:
The value of the given expression can be calculated step by step to find that it can be written as k * 12¹⁰⁰¹, where k = 4(3¹⁰⁰¹)(4¹⁰⁰²).
Step-by-step explanation:
The value of the expression (3¹⁰⁰¹+4¹⁰⁰²)²-(3¹⁰⁰¹-4¹⁰⁰²)² can be calculated step by step:
(3¹⁰⁰¹+4¹⁰⁰²)² = (3¹⁰⁰¹)² + 2(3¹⁰⁰¹)(4¹⁰⁰²) + (4¹⁰⁰²)²
(3¹⁰⁰¹-4¹⁰⁰²)² = (3¹⁰⁰¹)² - 2(3¹⁰⁰¹)(4¹⁰⁰²) + (4¹⁰⁰²)²
Substitute these values back into the main expression:
(3¹⁰⁰¹+4¹⁰⁰²)²-(3¹⁰⁰¹-4¹⁰⁰²)² = (3¹⁰⁰¹)² + 2(3¹⁰⁰¹)(4¹⁰⁰²) + (4¹⁰⁰²)² - (3¹⁰⁰¹)² + 2(3¹⁰⁰¹)(4¹⁰⁰²) - (4¹⁰⁰²)²
Simplify the expression:
(3¹⁰⁰¹+4¹⁰⁰²)²-(3¹⁰⁰¹-4¹⁰⁰²)² = 4(3¹⁰⁰¹)(4¹⁰⁰²)
So, the expression can be written as k * 12¹⁰⁰¹, where k = 4(3¹⁰⁰¹)(4¹⁰⁰²).