Question

Given α,β, respectively, the fifth and the fourth non-real roots of units, then find the value of

(1+α)(1+β)(1+α²)(1+β²)(1+α⁴)(1+β⁴)

Answer 1

To find the value of (1+α)(1+β)(1+α²)(1+β²)(1+α⁴)(1+β⁴), we can simplify the expression using the properties of complex numbers. This will result in the expression 2(1+α)(1+β)(1+α²)(1+β²).

Step-by-step explanation:

We can use the properties of complex numbers to find the value of (1+α)(1+β)(1+α²)(1+β²)(1+α⁴)(1+β⁴).

Since α and β are the fifth and fourth non-real roots of unity, we know that α⁵ = 1 and β⁴ = 1. Therefore, we can simplify the expression as follows:

(1+α)(1+β)(1+α²)(1+β²)(1+α⁴)(1+β⁴) = (1+α)(1+β)(1+α²)(1+β²)(1+1)(1+1)

Expanding the expression, we get:

(1+α)(1+β)(1+α²)(1+β²)(1+1)(1+1) = (1+α)(1+β)(1+α²)(1+β²)(2)(2)

Finally, we can multiply the terms together:

(1+α)(1+β)(1+α²)(1+β²)(2)(2) = 2(1+α)(1+β)(1+α²)(1+β²)