Understanding the Problem: Cube Roots of Unity

The question asks us to evaluate a complex expression involving \(\omega\), where \(\omega\) is a cube root of unity. We are given the expression \(\sqrt {\frac{{1 + {{\rm{\omega }}^2}}}{{1 + {\rm{\omega }}}}} \) and need to find its value among the given options.

A cube root of unity is a complex number \(z\) such that \(z^3 = 1\). The three cube roots of unity are 1, \(\omega\), and \(\omega^2\), where \(\omega\) is a non-real cube root of unity. These roots have specific properties that are crucial for solving this problem.

Key Properties of Cube Roots of Unity

For the non-real cube root of unity, \(\omega\), the following properties hold true:

• The sum of the roots is zero: \(1 + \omega + \omega^2 = 0\)
• The product of the roots is one: \(\omega^3 = 1\)

From the sum property, we can derive useful relations:

• \(1 + \omega^2 = -\omega\)
• \(1 + \omega = -\omega^2\)

Simplifying the Expression Step-by-Step

We need to simplify the expression \(\sqrt {\frac{{1 + {{\rm{\omega }}^2}}}{{1 + {\rm{\omega }}}}} \).

Let's first simplify the fraction inside the square root using the properties we just listed.

The fraction is \(\frac{{1 + {{\rm{\omega }}^2}}}{{1 + {\rm{\omega }}}}\).

Using the property \(1 + \omega^2 = -\omega\), we replace the numerator:

Numerator = \(-\omega\)

Using the property \(1 + \omega = -\omega^2\), we replace the denominator:

Denominator = \(-\omega^2\)

Now, substitute these into the fraction:

The fraction becomes \(\frac{{ - \omega }}{{ - \omega^2}}\)

Simplify the fraction:

\(\frac{{ - \omega }}{{ - \omega^2}} = \frac{\omega}{{\omega^2}}\)

We can simplify \(\frac{\omega}{{\omega^2}}\) using the property \(\omega^3 = 1\). From \(\omega^3 = 1\), we can divide both sides by \(\omega\) (since \(\omega \ne 0\)) to get \(\omega^2 = \frac{1}{\omega}\). Similarly, dividing by \(\omega^2\) gives \(\omega = \frac{1}{\omega^2}\).

So, \(\frac{\omega}{{\omega^2}} = \frac{1}{\omega}\). And since \(\frac{1}{\omega} = \omega^2\), the simplified fraction is \(\omega^2\).

Alternatively, \(\frac{\omega}{{\omega^2}} = \omega^{1-2} = \omega^{-1}\). Since \(\omega^3=1\), \(\omega^{-1} = \omega^{-1} \cdot \omega^3 = \omega^{-1+3} = \omega^2\).

So, the fraction \(\frac{{1 + {{\rm{\omega }}^2}}}{{1 + {\rm{\omega }}}}} \) simplifies to \(\omega^2\).

Now, we need to find the square root of this simplified fraction:

\(\sqrt {\frac{{1 + {{\rm{\omega }}^2}}}{{1 + {\rm{\omega }}}}} = \sqrt {{\omega ^2}} \)

The expression \(\sqrt{{\omega^2}}\) represents the complex numbers whose square is equal to \(\omega^2\). The numbers that satisfy this condition are \(\omega\) and \(-\omega\).

We are given four options for the value of the expression:

1. 1
2. \(\omega\)
3. \(\omega^2\)
4. \(i\omega\)

Comparing the possible values (\(\omega\) and \(-\omega\)) with the given options, we see that \(\omega\) is one of the options.

Therefore, the value of the expression is \(\omega\).

Revision Table: Cube Root Properties Applied

Additional Information: Understanding Cube Roots of Unity

The cube roots of unity are the solutions to the equation \(z^3 = 1\). In the complex plane, these roots are located on the unit circle at angles \(0\), \(2\pi/3\), and \(4\pi/3\) radians from the positive real axis.

• The root at angle 0 is \(e^{i0} = \cos(0) + i\sin(0) = 1\).
• The root at angle \(2\pi/3\) is \(\omega = e^{i2\pi/3} = \cos(2\pi/3) + i\sin(2\pi/3) = -\frac{1}{2} + i\frac{\sqrt{3}}{2}\).
• The root at angle \(4\pi/3\) is \(\omega^2 = e^{i4\pi/3} = \cos(4\pi/3) + i\sin(4\pi/3) = -\frac{1}{2} - i\frac{\sqrt{3}}{2}\).

Notice that \(\omega^2\) is also the complex conjugate of \(\omega\), denoted as \(\bar{\omega}\).

The properties \(1 + \omega + \omega^2 = 0\) and \(\omega^3 = 1\) are fundamental and frequently used in problems involving cube roots of unity in complex numbers.