We are given the following system of equations:

$a + b + c = 4$ $a^2 + b^2 + c^2 = 10$ $a^3 + b^3 + c^3 = 22$ We are tasked with finding the value of $a^4 + b^4 + c^4$.

Step 1: Use Elementary Symmetric Polynomials

We start by recognizing that these equations can be handled using symmetric polynomials. Define:

• $p_1 = a + b + c$
• $p_2 = ab + bc + ca$
• $p_3 = abc$

Given:

• $p_1 = 4$

We will use the relations for power sums: $a^2 + b^2 + c^2 = p_1^2 - 2p_2$ Substituting the known values: $10 = 4^2 - 2p_2$ $10 = 16 - 2p_2$ Solving for $p_2$: $p_2 = 3$

Step 2: Use the Third Equation

We can express $a^3 + b^3 + c^3$ in terms of $p_1, p_2, p_3$: $a^3 + b^3 + c^3 = p_1^3 - 3p_1p_2 + 3p_3$ Substitute the known values: $22 = 4^3 - 3(4)(3) + 3p_3$ $22 = 64 - 36 + 3p_3$ $22 = 28 + 3p_3$ Solving for $p_3$: $p_3 = -2$

Step 3: Find $a^4 + b^4 + c^4$

We can express $a^4 + b^4 + c^4$ as: $a^4 + b^4 + c^4 = (a^2 + b^2 + c^2)^2 - 2(a^2b^2 + b^2c^2 + c^2a^2)$ We already know that $a^2 + b^2 + c^2 = 10$, so: $a^4 + b^4 + c^4 = 10^2 - 2(a^2b^2 + b^2c^2 + c^2a^2)$ Now, we calculate $a^2b^2 + b^2c^2 + c^2a^2$ using: $a^2b^2 + b^2c^2 + c^2a^2 = p_2^2 - 2p_1p_3$ Substitute the known values: $a^2b^2 + b^2c^2 + c^2a^2 = 3^2 - 2(4)(-2)$ $a^2b^2 + b^2c^2 + c^2a^2 = 9 + 16 = 25$

Finally, substitute this into the expression for $a^4 + b^4 + c^4$: $a^4 + b^4 + c^4 = 100 - 2(25) = 100 - 50 = 50$

Thus, the value of $a^4 + b^4 + c^4$ is $\boxed{50}$.

Would you like further details on any step? Here are some related questions to consider:

1. How can the method of symmetric polynomials be applied to higher-degree equations?
2. What is the general form of the power sum symmetric polynomials?
3. Could these results help in solving cubic or quartic equations?
4. What other techniques can solve similar systems of equations?
5. How would the approach differ if the equations were non-symmetric?

Tip: In problems involving powers of variables and sums, symmetric polynomials simplify the work by reducing the number of variables and focusing on key invariants.