1. **State the problem:** Given three variables $a$, $b$, and $c$ with the following sums: $$a + b + c = 4$$ $$a^2 + b^2 + c^2 = 10$$ $$a^3 + b^3 + c^3 = 22$$ Find the value of: $$a^4 + b^4 + c^4$$ 2. **Recall the formula and rules:** We can use Newton's identities which relate power sums of roots to the elementary symmetric sums. Define: $$p_k = a^k + b^k + c^k$$ and the elementary symmetric sums: $$e_1 = a + b + c$$ $$e_2 = ab + bc + ca$$ $$e_3 = abc$$ Newton's identities for three variables state: $$p_1 = e_1$$ $$p_2 = e_1 p_1 - 2 e_2$$ $$p_3 = e_1 p_2 - e_2 p_1 + 3 e_3$$ $$p_4 = e_1 p_3 - e_2 p_2 + e_3 p_1$$ 3. **Find $e_2$ and $e_3$ using given data:** From $p_1 = e_1 = 4$ Use $p_2 = e_1 p_1 - 2 e_2$: $$10 = 4 \times 4 - 2 e_2$$ $$10 = 16 - 2 e_2$$ $$2 e_2 = 16 - 10 = 6$$ $$e_2 = 3$$ Use $p_3 = e_1 p_2 - e_2 p_1 + 3 e_3$: $$22 = 4 \times 10 - 3 \times 4 + 3 e_3$$ $$22 = 40 - 12 + 3 e_3$$ $$22 = 28 + 3 e_3$$ $$3 e_3 = 22 - 28 = -6$$ $$e_3 = -2$$ 4. **Calculate $p_4$ using Newton's identity:** $$p_4 = e_1 p_3 - e_2 p_2 + e_3 p_1$$ Substitute known values: $$p_4 = 4 \times 22 - 3 \times 10 + (-2) \times 4$$ $$p_4 = 88 - 30 - 8$$ $$p_4 = 50$$ **Final answer:** $$a^4 + b^4 + c^4 = 50$$