If 1, π,π 2 are cube roots of unity, show that (2 – π) (2 – π 2 ) (2 – π 19) (2 – π 23) = 49.

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The cube roots of unity are given by:

$1, \omega, \omega^2$

where $$\omega = e^{2\pi i/3}$$ and $$\omega^2 = e^{4\pi i/3}$$.

Let's find the expression $$(2 - \omega)(2 - \omega^2)(2 - \omega^{19})(2 - \omega^{23})$$:

$(2 - \omega)(2 - \omega^2)(2 - \omega^{19})(2 - \omega^{23})$

Using the fact that $$\omega^3 = 1$$, we can simplify this expression. First, consider the terms $$(2 - \omega)(2 - \omega^2)$$:

$(2 - \omega)(2 - \omega^2) = 4 - 2(\omega + \omega^2) + \omega \omega^2$

Since $$\omega^2 = 1 - \omega$$, we can substitute this into the expression:

$4 - 2(\omega + 1 - \omega) + \omega(1 - \omega)$

Now, simplify further:

$4 - 2\omega - 2 + \omega - \omega^2$

Since $$\omega^2 = 1 - \omega$$, substitute this in:

$4 - 2\omega - 2 + \omega - (1 - \omega)$

Combine like terms:

$1 - 2\omega$

Now, consider the product $$(2 - \omega^{19})(2 - \omega^{23})$$. Since $$\omega^3 = 1$$, we can write $$\omega^{19}$$ and $$\omega^{23}$$ as $$\omega^{3(6) + 1}$$ and $$\omega^{3(7) + 2}$$, respectively:

$(2 - \omega^{19})(2 - \omega^{23}) = (2 - \omega) (2 - \omega^2)$

We've already found that this expression simplifies to $$1 - 2\omega$$.

Now, combine the results:

$(2 - \omega)(2 - \omega^2)(2 - \omega^{19})(2 - \omega^{23}) = (1 - 2\omega)(1 - 2\omega)$

$= 1 - 4\omega + 4\omega^2$

Using the fact that $$\omega^2 = 1 - \omega$$, substitute this in:

$= 1 - 4\omega + 4(1 - \omega)$

$= 1 - 4\omega + 4 - 4\omega$

Combine like terms:

$= 5 - 8\omega$

Now, substitute the values of $$\omega$$ and $$\omega^2$$:

$= 5 - 8e^{2\pi i/3}$

$= 5 - 8\left(-\frac{1}{2} + \frac{\sqrt{3}}{2}i\right)$

$= 5 + 4 - 4\sqrt{3}i$

$= 9 - 4\sqrt{3}i$

Now, multiply the complex conjugate to eliminate the imaginary part:

$(9 - 4\sqrt{3}i)(9 + 4\sqrt{3}i)$

Using the difference of squares formula $$(a - b)(a + b) = a^2 - b^2$$:

$= 81 - (4\sqrt{3})^2$

$= 81 - 48$

$= 33$

Therefore, $$(2 - \omega)(2 - \omega^2)(2 - \omega^{19})(2 - \omega^{23}) = 33$$.