Problem 172 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The values of \((16)^{1 / 4}\) are (A) \(\pm 2, \pm 2 i\) (B) \(\pm 4, \pm 4 i\) (C) \(\pm 1, \pm i\) (D) None of these

Step-by-Step Solution

Answer

The correct answer is (A) \(\pm 2, \pm 2i\).

Step 1: Understand the Problem

We need to determine the fourth roots of 16, written as \((16)^{1/4}\), and identify which set of values includes all possible roots. The fourth root of a number can include both real and complex numbers.

Step 2: Convert into Exponential Form

Express 16 in exponential form. Since 16 is a power of 2, we can write it as \(16 = 2^4\). Hence, \((16)^{1/4} = (2^4)^{1/4}\).

Step 3: Apply the Power of a Power Rule

Using the rule \((a^m)^n = a^{m \cdot n}\), we have \((2^4)^{1/4} = 2^{4 \cdot (1/4)}\). Simplifying gives us \(2^1 = 2\). Thus, \(2\) is one of the roots.

Step 4: Consider Complex Roots

Since we are looking for all fourth roots, we must include complex numbers. The fourth roots of a number \(c\), when considering its exponential form in the complex plane, are given by \(c^{1/4} \cdot e^{i\frac{(2k\pi)}{4}}\) for \(k = 0, 1, 2, 3\).

Step 5: Calculate All Roots Using Euler's Formula

Using Euler's formula, find the other roots: \((16)^{1/4} = 16^{1/4} e^{i\frac{(2\pi k)}{4}} = 2e^{i\frac{\pi k}{2}}\).1. For \(k = 0\), root is \(2e^{i \cdot 0} = 2\).2. For \(k = 1\), root is \(2e^{i \cdot \frac{\pi}{2}} = 2i\).3. For \(k = 2\), root is \(2e^{i \cdot \pi} = -2\).4. For \(k = 3\), root is \(2e^{i \cdot \frac{3\pi}{2}} = -2i\).

Step 6: Compare with Given Options

The values we obtained are \(\pm 2, \pm 2i\). Check the answer choices. The correct answer is (A), which matches our calculated roots: \(\pm 2, \pm 2i\).

Key Concepts

Roots of Unity

Roots of unity are special complex numbers that represent the possible distinct solutions to the equation \(x^n = 1\). These roots are essential in understanding complex numbers and have unique geometrical meanings on the unit circle in the complex plane. For example, if you are finding the fourth roots of unity, you're solving for numbers \(z\) such that \(z^4 = 1\).Let's explore how these concepts apply:

• The unit circle in the complex plane has a radius of 1 and center at the origin.
• Each root of unity lies on this circle.
• For the fourth roots of unity, you will have four separate roots, equally distributed around the circle.

The roots are given by the formula: \(e^{i \frac{2k\pi}{n}}\), where \(k\) ranges from 0 to \(n-1\), and \(n\) is the degree of the root. This formula is a direct consequence of the rotational symmetry of the complex plane.

Exponential Form

The exponential form of complex numbers is a very useful way to handle operations like multiplication and finding roots. It represents complex numbers in the form \(re^{i\theta}\), where \(r\) is the magnitude and \(\theta\) is the argument or angle with the positive real axis. This form makes it easier to understand and compute operations involving powers and roots.Here's a breakdown of the usage:

• Convert real numbers to complex numbers by viewing them as having an imaginary part of 0.
• Express any number's root by finding its exponential expression first, which allows the application of complex operations.

By expressing 16 in exponential form as \(2^4\), finding \((16)^{1/4}\) boils down to finding the roots of \(2\) in the exponential form, making the task much simpler. It leverages the simplicity of multiplication and division in the exponential realm.

Euler's Formula

Euler's Formula is a key element in understanding complex numbers, providing a bridge between exponential and trigonometric functions. It is stated as \(e^{i\theta} = \cos{\theta} + i\sin{\theta}\). This formula enables the representation of complex numbers on the unit circle, effectively linking angle and radius.Here's what you need to know:

• The formula helps convert between exponential and trigonometric forms easily.
• It shows that complex numbers can be represented as rotations around the origin in the complex plane.
• Using Euler's formula simplifies the calculation of roots of complex numbers as it connects the function \(e^x\) directly to the circular functions \(\cos\) and \(\sin\).

In solving our problem, Euler's Formula is used to compute \((16)^{1/4} = 2e^{i\frac{\pi k}{2}}\). By adjusting the parameter \(k\), we can systematically find all distinct roots, covering all possible rotations by increments of \(90^\circ\) (or \(\pi/2\) radians). Thus, Euler's Formula simplifies the pursuit of complex roots by transforming the calculations into manageable pieces.