If 1, 𝜔,𝜔 2 are cube roots of unity, show that (2 – 𝜔) (2 – 𝜔 2 ) (2 – 𝜔 19) (2 – 𝜔 23) = 49.

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