Hardest maths questions - find a^4 + b^4 + c^4 - Free Printable

We are given the following system of equations:

1. \( a + b + c = 4 \)
2. \( a^2 + b^2 + c^2 = 10 \)
3. \( a^3 + b^3 + c^3 = 22 \)

We need to find the value of \( a^4 + b^4 + c^4 \).

Step 1: Use the identity for the sum of squares

The identity for the sum of squares is:
\[
a^2 + b^2 + c^2 = (a + b + c)^2 - 2(ab + bc + ca)
\]
Substitute the given values:
\[
10 = 4^2 - 2(ab + bc + ca)
\]
\[
10 = 16 - 2(ab + bc + ca)
\]
Solve for \( ab + bc + ca \):
\[
2(ab + bc + ca) = 16 - 10
\]
\[
2(ab + bc + ca) = 6
\]
\[
ab + bc + ca = 3
\]

Step 2: Use the identity for the sum of cubes

The identity for the sum of cubes is:
\[
a^3 + b^3 + c^3 = (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca) + 3abc
\]
Substitute the known values:
\[
22 = 4(10 - 3) + 3abc
\]
\[
22 = 4 \cdot 7 + 3abc
\]
\[
22 = 28 + 3abc
\]
Solve for \( abc \):
\[
3abc = 22 - 28
\]
\[
3abc = -6
\]
\[
abc = -2
\]

Step 3: Use the identity for the sum of fourth powers

The identity for the sum of fourth powers is:
\[
a^4 + b^4 + c^4 = (a^2 + b^2 + c^2)^2 - 2(a^2b^2 + b^2c^2 + c^2a^2)
\]
We already know \( a^2 + b^2 + c^2 = 10 \). Now, we need to find \( a^2b^2 + b^2c^2 + c^2a^2 \).

Using the identity:
\[
(a^2b^2 + b^2c^2 + c^2a^2) = (ab + bc + ca)^2 - 2abc(a + b + c)
\]
Substitute the known values:
\[
(ab + bc + ca)^2 = 3^2 = 9
\]
\[
abc(a + b + c) = (-2)(4) = -8
\]
\[
a^2b^2 + b^2c^2 + c^2a^2 = 9 - 2(-8)
\]
\[
a^2b^2 + b^2c^2 + c^2a^2 = 9 + 16
\]
\[
a^2b^2 + b^2c^2 + c^2a^2 = 25
\]

Now substitute back into the identity for \( a^4 + b^4 + c^4 \):
\[
a^4 + b^4 + c^4 = (a^2 + b^2 + c^2)^2 - 2(a^2b^2 + b^2c^2 + c^2a^2)
\]
\[
a^4 + b^4 + c^4 = 10^2 - 2 \cdot 25
\]
\[
a^4 + b^4 + c^4 = 100 - 50
\]
\[
a^4 + b^4 + c^4 = 50
\]

Thus, the value of \( a^4 + b^4 + c^4 \) is:
\[
\boxed{50}
\]