First, recognize that $\alpha$ and $\beta$ are cube roots of unity. Specifically, $\alpha$ and $\beta$ satisfy the equation $x^2 + x + 1 = 0$, since one of their sum is $-1$ and their product is 1. In terms of cube roots of unity, $\alpha$ is $\omega$ and $\beta$ is $\omega^2$, where $\omega = e^{2\pi i/3}$. These satisfy $\omega^3 = 1$ and $1 + \omega + \omega^2 = 0$.
Next, compute:
- For $A = 7 - 7\alpha + 9\beta$:
- $7 - 7\alpha + 9\beta = 7 - 7\omega + 9\omega^2$
- Using $1 + \omega + \omega^2 = 0$, rewrite $7 - 7(\omega - 1) + 9(\omega^2 - 1) = 7 + 2 = 21$
- For $B = 9 + 7\alpha - 7\beta$:
- $9 + 7\omega - 7\omega^2 = 9 + 7(\omega - \omega^2) \to 9 + 7(-1) = 2$
- For $C = -7 + 9\alpha + 7\beta$:
- $-7 + 9\omega + 7\omega^2 = -7 + 9(-\omega^2 - 1) + 7$ from earlier
- Thus, rewrite as $-7 + 16 = 9
- For $D = 14 + 7\alpha + 7\beta$:
- $14 + 7\omega + 7\omega^2 = 14 + 7(-1) = 7
Compute the sum of each computed value to the power of 20:
$A^{20} + B^{20} + C^{20} + D^{20} = 21^{20} + 2^{20} + 9^{20} + 7^{20}$. Recognize complex terms are unit root rotations changing cyclically every third power multiplied with unity j-th roots characters, rendering computations inefficiently large, simplifying down to integers solely influenced by unity roots, concluding each path internally rooted as 1 or sum $(1)^{20}$.
This expression equated to $m^{10}$ simplifies naturally to overall unity schema, reaching an LCM effector of 49 unit harmonic multipliers.
Similar Questions
- Let A be a $3 \times 3$ matrix such that $A + A^T = O$. If $A\begin{bmatrix} 1 \\ -1 \\ 0 \end{bmatrix} = \begin{bmatrix} 3 \\ 3 \\ 2 \end{bmatrix}$, $A^2\begin{bmatrix} 1 \\ -1 \\ 0 \end{bmatrix} = \begin{bmatrix} -3 \\ 19 \\ -24 \end{bmatrix}$ and $\det(adj(2 \ adj(A + I))) = (2)^\alpha \cdot (3)^\beta \cdot (11)^\gamma$, $\alpha, \beta, \gamma$ are non-negative integers, then $\alpha + \beta + \gamma$ is equal to _________
- Let ABC be a triangle. Consider four points $p_1, p_2, p_3, p_4$ on the side AB, five points $p_5, p_6, p_7, p_8, p_9$ on the side BC, and four points $p_{10}, p_{11}, p_{12}, p_{13}$ on the side AC. None of these points is a vertex of the triangle ABC. Then the total number of pentagons, that can be formed by taking all the vertices from the points $p_1, p_2, ..., p_{13}$, is _________
- Let n be the number obtained on rolling a fair die. If the probability that the system
$x - ny + z = 6$
$x + (n - 2)y + (n + 1)z = 8$
$(n - 1)y + z = 1$
has a unique solution is $\frac{k}{6}$, then the sum of k and all possible values of n is : - If $X = \begin{bmatrix} x \\ y \\ z \end{bmatrix}$ is a solution of the system of equations $AX = B$, where $\text{adj } A = \begin{bmatrix} 4 & 2 & 2 \\ -5 & 0 & 5 \\ 1 & -2 & 3 \end{bmatrix}$ and $B = \begin{bmatrix} 4 \\ 0 \\ 2 \end{bmatrix}$, then $|x + y + z|$ is equal to :
- Let $\alpha, \beta$ be the roots of the quadratic equation $12x^2 - 20x + 3\lambda = 0, \lambda \in \mathbb{Z}$. If $\frac{1}{2} \leq |\beta - \alpha| \leq \frac{3}{2}$, then the sum of all possible values of $\lambda$ is :
- Let $C_r$ denote the coefficient of $x^r$ in the binomial expansion of $(1 + x)^n$, $n \in \mathbb{N}, 0 \leq r \leq n$. If $P_n = C_0 - C_1 + \frac{2^2}{3} C_2 - \frac{2^3}{4} C_3 + \dots + \frac{(-2)^n}{n+1} C_n$, then the value of $\sum_{n=1}^{25} \frac{1}{P_{2n}}$ equals.
- The number of elements in the relation $R = \{(x, y) : 4x^2 + y^2 < 52, x, y \in \mathbb{Z}\}$ is
- Let $S = \{z \in \mathbb{C} : 4z^2 + \bar{z} = 0\}$. Then $\sum_{z \in S} |z|^2$ is equal to :
Let f and g be functions satisfying $f(x+y) = f(x)f(y), f(1) = 7$ and $g(x+y) = g(xy), g(1) = 1$, for all $x, y \in \mathbb{N}$. If $\sum_{x=1}^{n} \left(\frac{f(x)}{g(x)}\right) = 19607$, then n is equal to :
- Suppose a, b, c are in A.P. and $a^2, 2b^2, c^2$ are in G.P. If $a < b < c$ and $a + b + c = 1$, then $9(a^2 + b^2 + c^2)$ is equal to ________.
Important Questions from Algebra
- Let A be a $3 \times 3$ matrix such that $A + A^T = O$. If $A\begin{bmatrix} 1 \\ -1 \\ 0 \end{bmatrix} = \begin{bmatrix} 3 \\ 3 \\ 2 \end{bmatrix}$, $A^2\begin{bmatrix} 1 \\ -1 \\ 0 \end{bmatrix} = \begin{bmatrix} -3 \\ 19 \\ -24 \end{bmatrix}$ and $\det(adj(2 \ adj(A + I))) = (2)^\alpha \cdot (3)^\beta \cdot (11)^\gamma$, $\alpha, \beta, \gamma$ are non-negative integers, then $\alpha + \beta + \gamma$ is equal to _________
- Let ABC be a triangle. Consider four points $p_1, p_2, p_3, p_4$ on the side AB, five points $p_5, p_6, p_7, p_8, p_9$ on the side BC, and four points $p_{10}, p_{11}, p_{12}, p_{13}$ on the side AC. None of these points is a vertex of the triangle ABC. Then the total number of pentagons, that can be formed by taking all the vertices from the points $p_1, p_2, ..., p_{13}$, is _________
- Let n be the number obtained on rolling a fair die. If the probability that the system
$x - ny + z = 6$
$x + (n - 2)y + (n + 1)z = 8$
$(n - 1)y + z = 1$
has a unique solution is $\frac{k}{6}$, then the sum of k and all possible values of n is : - If $X = \begin{bmatrix} x \\ y \\ z \end{bmatrix}$ is a solution of the system of equations $AX = B$, where $\text{adj } A = \begin{bmatrix} 4 & 2 & 2 \\ -5 & 0 & 5 \\ 1 & -2 & 3 \end{bmatrix}$ and $B = \begin{bmatrix} 4 \\ 0 \\ 2 \end{bmatrix}$, then $|x + y + z|$ is equal to :
- Let $\alpha, \beta$ be the roots of the quadratic equation $12x^2 - 20x + 3\lambda = 0, \lambda \in \mathbb{Z}$. If $\frac{1}{2} \leq |\beta - \alpha| \leq \frac{3}{2}$, then the sum of all possible values of $\lambda$ is :
