Question

$\huge\bf{\red{\underline{\underline{{QueStion}}}}}$

if α, β, γ are such that:-
α+β+γ = 2, α²+β²+γ² = 6 and α³+β³+γ³ = 8, then α⁴+β⁴+γ⁴ = ?

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Answer 1

Answer:

(α+β+γ)2= α2+ β2+ γ2

+2(αβ+βγ+γα)⇒αβ+βγ+γα=−1

And α3+ β3+ γ3

−3αβγ=(α+β+γ)(α2+β2+γ2−αβ−βγ−γα)

⇒αβγ=−2

Then

(α2 +β2 +γ2 )2

=∑α4+2∑β2 γ2

=∑α4+2((∑βγ)2−2αβγ(α+β+γ))

⇒α4 +β4+ γ4

=36−2((−1)2−2(−2)2)

=18

Answer 2

Answer:

$\alpha^4 + \beta^4 + \gamma^4 = 18$

Step-by-step explanation:

The most direct, although also an extremely tedious way of doing this is through the normal way of isolating variables and solving the equations, as you would normally do with a system of equations.

This is however, one of those problems that can be quickly solved by using a few tricks and trial and error.

First thing we notice is that all the equations are symmetric in $\alpha, \beta, \gamma$, that is, we can freely exchange the values of the variables and the equations remain true.

Secondly, we notice that if we raised all variables to the zeroth power, $\alpha^0 + \beta^0 +\gamma^0$, we would get 3, which is less than 2. This implies that at least one of the variables are less than 1.

We also notice that the growth pattern is not exponential as changing the exponent from 1 to 2 increases the value by 4, while increasing the exponent from 2 to 3 increases the value by 2. This is a decrease in growth, not an increase, contrary to what we would expect from exponential growth. This means that at least one of the variables must be negative.

Looking at $\alpha + \beta + \gamma = 2$,  a natural guess would be $\alpha = 2, \beta = -\gamma$. Then the second equation becomes: $2^2 + (-\gamma)^2 + \gamma ^2 = 6$. This then gives:

$2\gamma^2 = 2 \iff \gamma^2 = 1 \iff \gamma = \pm 1$.

As we said before, the equations are symmetric in their variables so we can simply pick $\gamma = 1$, giving $\beta = -1$.

Inputing the values into the third equation gives:

$2^3 + (-1)^3 + 1^3 = 8 -1 + 1 = 6$, which is the right value. Hence our choices for $\alpha, \beta, \gamma$ satisfy the equations.

We can then compute the desired value:

$2^4 + (-1)^4 + 1^4 = 16 +1 +1 = 18.$