If  α, β be two roots of the equation x2 + (24)1/4x + 61/2 = 0, then  α8 + β8 is equal to

Let's solve the problem step by step:

Step 1: Given equation The given equation is x^2 + (24)^(1/4)x + 6^(1/2) = 0.

Step 2: Find the sum and product of the roots Using the sum and product of roots formula, we can find the sum and product of the roots: Sum of roots (α + β) = -b/a = -((24)^(1/4))/1 = -(24)^(1/4) Product of roots (αβ) = c/a = (6)^(1/2)/1 = (6)^(1/2)

Step 3: Find α^8 + β^8 To find α^8 + β^8, we need to find the value of α^8 and β^8 individually.

Step 4: Finding α^8 We can express α^8 in terms of the sum and product of roots using the binomial expansion: α^8 = (α^2)^4 = (α^2)^2 * (α^2)^2 = (α^2 + β^2)^2

Step 5: Finding α^2 + β^2 Using the sum and product of roots formula, we can find α^2 + β^2: (α^2 + β^2) = (α + β)^2 - 2αβ = ((24)^(1/4))^2 - 2(6)^(1/2) = 24^(1/2) - 2(6)^(1/2)

Step 6: Substitute the value of α^2 + β^2 in α^8 Substituting the value of α^2 + β^2 in α^8, we get: α^8 = (24^(1/2) - 2(6)^(1/2))^2

Step 7: Simplify α^8 Expanding and simplifying α^8, we get: α^8 = 24 - 4(6)^(1/2) + 4(6)^(1/2) - 24 = 0

Step 8: Finding β^8 Using the same steps as above, we can find β^8: β^8 = (24^(1/2) - 2(6)^(1/2))^2 = 0

Step 9: Finding α^8 + β^8 Finally, we can find α^8 + β^8 by adding the values of α^8 and β^8: α^8 + β^8 = 0 + 0 = 0

Therefore, α^8 + β^8 is equal to 0.