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We are given the equations:

a+b=4a + b = 4 a2+b2=6a^2 + b^2 = 6

We aim to determine the value of a3+b3a^3 + b^3.

Step 1: Express a2+b2a^2 + b^2 Using a+ba + b

Using the identity:

a2+b2=(a+b)2−2aba^2 + b^2 = (a+b)^2 - 2ab

Substituting the known values:

6=42−2ab6 = 4^2 - 2ab 6=16−2ab6 = 16 - 2ab 2ab=102ab = 10 ab=5ab = 5

Step 2: Compute a3+b3a^3 + b^3

Using the identity:

a3+b3=(a+b)(a2−ab+b2)a^3 + b^3 = (a+b)(a^2 - ab + b^2)

Since we already know a+b=4a+b = 4, a2+b2=6a^2 + b^2 = 6, and ab=5ab = 5, we compute:

a2−ab+b2=6−5=1a^2 - ab + b^2 = 6 - 5 = 1

Thus,

a3+b3=4×1=4a^3 + b^3 = 4 \times 1 = 4

Final Answer:

{4}