Problem 193 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(\omega\) is an imaginary cube root of unity, then \((1+\omega\) \(\left.-\omega^{2}\right)^{7}\) equals: (A) \(128 \omega\) (B) \(-128 \omega\) (C) \(128 \omega^{2}\) (D) \(-128 \omega^{2}\)

Step-by-Step Solution

Answer

The correct answer is (B) \\-128 \\omega\\.

Step 1: Understanding Imaginary Cube Roots

The imaginary cube roots of unity are \( ext{1, } \omega, \omega^{2} \), where \( ext{1 + } \omega + \omega^{2} = 0\). It's also important to remember that \( \omega^{3} = 1 \), and its cyclic property implies \( ext{ powers of } \omega \text{ reduce to either } \omega \text{ or } \omega^{2} \text{ or 1.}\)

Step 1: Simplify Inside the Parentheses

Start simplifying the expression \(1 + \omega - \omega^{2}\). By rewriting using the known property that \(1 + \omega + \omega^{2} = 0\), we can replace \(1 = - \omega - \omega^{2}\). Substitute to get \(- \omega - \omega^{2} + \omega - \omega^{2}\). The expression simplifies to \(-2 \omega^{2}\).

Step 2: Calculate the Power of the Simplified Expression

Now, calculate \((-2 \omega^{2})^{7}\). This becomes \((-2)^7 imes (\omega^{2})^7\). Since \( \omega^{3} = 1 \), \( \omega^{6} = 1 \), and thus \( \omega^{7} = \omega\). \((-2)^7 = -128\). This gives the result of \(-128 \omega\).

Step 3: Select the Correct Answer

From the options provided, the expression simplifies to \(-128 \omega\). Therefore, option (B) \-128 \omega\ is the correct answer.

Key Concepts

Cube Roots of Unity

The cube roots of unity are fascinating elements in complex number theory. They have specific properties that make manipulating them quite elegant. The cube roots of unity consist of the numbers: 1, \(\omega\), and \(\omega^2\). These satisfy the equation \(x^3 = 1\).

The value \(\omega\) is an essential constant here, representing a complex number other than 1, but satisfying the same equation. The key properties of these roots are:

• All three roots sum to zero: \(1 + \omega + \omega^2 = 0\).
• Multiplying them in any sequence yields one: \(\omega \cdot \omega^2 \cdot 1 = 1\).
• Each has a magnitude of one on the complex plane.

These properties are not only useful for solving equations involving complex numbers, but they also help in understanding polynomial roots and transformations in the complex plane.

Exponentiation of Complex Numbers

Raising complex numbers to a power involves both angles and magnitudes. Specifically, with cube roots like \(\omega\), the properties simplify these calculations. Consider \((-2 \omega^2)^7\), where you need to raise both the magnitude and the complex part to the power of seven.

Breaking it down,

• Magnitude: \((-2)^7 = -128\)
• Exponentiation of \(\omega^2\): Using \(\omega^3 = 1\), we note \((\omega^2)^7 = \omega^{14}\). Since \(\omega^{3} = 1\), \(\omega^{6} = 1\), simplifying further implies \(\omega^{14} = \omega^{14 \mod 3} = \omega^2 \).

This step highlights the cyclic nature and periodicity inherent in complex roots, where the exponentiation is often reduced to simpler forms using modular arithmetic.

Simplification of Expressions

Simplifying expressions with complex numbers like \(1 + \omega - \omega^2\) involves substituting known identities. Using the cube root of unity identity \(1 + \omega + \omega^2 = 0\), this can be transformed for easier handling. Here’s how it unfolds:

• Begin by recognizing that \(1 = - \omega - \omega^2\).
• Substitute into the expression to get \(- \omega - \omega^2 + \omega - \omega^2\).
• The resulting calculation simplifies directly to \(-2\omega^2\).

Notice that identifying and substituting these identities efficiently reduces the complexity of the expression. This simplification is crucial for making problems more manageable, especially before exponentiating them.