Final answer:
To find the value of a⁴ + a⁻⁴, substitute the value of a into the expression a⁴ + (4 - a)⁻⁴ and simplify the equation.
Step-by-step explanation:
To find the value of a⁴ + a⁻⁴, we can use the equation 4 = a + a⁻¹. We are given that a satisfies this equation, so we can substitute the value of a into the expression. From the equation, we know that a + a⁻¹ = 4, which means that a⁻¹ = 4 - a. Now, let's substitute a⁻¹ into the expression a⁴ + a⁻⁴:
a⁴ + a⁻⁴ = a⁴ + (4 - a)⁻⁴ = a⁴ + (4 - a)⁴ = a⁴ + (4 - a)(4 - a)²
Expanding the expression, we get:
a⁴ + a⁻⁴ = a⁴ + 4(4 - a)³ - 6(4 - a)² + 4(4 - a) - 1
Simplifying further, we find:
a⁴ + a⁻⁴ = 166a⁴ - 2128a³ + 9856a² - 17984a + 10242
The value of a⁴ + a⁻⁴ is therefore 166a⁴ - 2128a³ + 9856a² - 17984a + 10242.