The image presents the following system of equations involving three variables $a$, $b$, and $c$:

a + b + c = 4 \tag{1} a^2 + b^2 + c^2 = 10 \tag{2} a^3 + b^3 + c^3 = 22 \tag{3}

You are asked to find the value of:

$a^4 + b^4 + c^4 = ?$

Step-by-step Solution:

1. Use the identity for sums of squares:

From Equation (1): $(a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca)$ Substitute the known values from (1) and (2): $4^2 = 10 + 2(ab + bc + ca)$ $16 = 10 + 2(ab + bc + ca)$ $6 = 2(ab + bc + ca)$ ab + bc + ca = 3 \tag{4}

2. Use the identity for sums of cubes:

Using the identity: $a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca)$ Substitute the known values from (1), (2), and (4): $22 - 3abc = 4(10 - 3)$ $22 - 3abc = 4 \times 7$ $22 - 3abc = 28$ $-3abc = 6$ abc = -2 \tag{5}

3. Calculate $a^4 + b^4 + c^4$:

Now, to find $a^4 + b^4 + c^4$, use the identity: $a^4 + b^4 + c^4 = (a^2 + b^2 + c^2)^2 - 2(a^2b^2 + b^2c^2 + c^2a^2)$ We already know $a^2 + b^2 + c^2 = 10$, so: $a^4 + b^4 + c^4 = 10^2 - 2(a^2b^2 + b^2c^2 + c^2a^2)$ To find $a^2b^2 + b^2c^2 + c^2a^2$, we use: $(ab + bc + ca)^2 = a^2b^2 + b^2c^2 + c^2a^2 + 2abc(a + b + c)$ Substitute from (4) and (5): $3^2 = a^2b^2 + b^2c^2 + c^2a^2 + 2(-2)(4)$ $9 = a^2b^2 + b^2c^2 + c^2a^2 - 16$ $a^2b^2 + b^2c^2 + c^2a^2 = 25$

Now substitute back: $a^4 + b^4 + c^4 = 100 - 2 \times 25$ $a^4 + b^4 + c^4 = 100 - 50$ $a^4 + b^4 + c^4 = 50$

Final Answer:

$a^4 + b^4 + c^4 = 50$

Would you like more details on any step or have any questions?

Here are five follow-up questions:

1. Can you apply a similar method to calculate higher powers of $a^5 + b^5 + c^5$?
2. How can the symmetric nature of equations help simplify calculations in other contexts?
3. What is the importance of identities like sums of squares and cubes in algebraic problems?
4. Can we generalize this approach for larger systems with more variables?
5. How would the problem change if one of the values of $a$, $b$, or $c$ was known?

Tip: Identifying symmetry in equations often simplifies the problem-solving process by reducing the number of variables or steps required.