The value of (1+i)^8-(1-i)^8 is (a) 0 (b) 16 (c) 32 (d) √(2)(1+i)

Name: Problem 102, Chapter 1: IIT JEE Super Course in Mathematics ALGEBRA II
Uploaded: 2021-06-05T14:51:29-08:00
Duration: 2 min
Channel: Anas Venkitta
Description: The value of (1+i)^8-(1-i)^8 is (a) 0 (b) 16 (c) 32 (d) √(2)(1+i)

step 1 So in the given question we are told to find the value of an expression that is given as 1 plus

The value of (1+i)^8-(1-i)^8 is (a) 0 (b) 16 (c) 32 (d)...

Key Concepts

De Moivre's Theorem

De Moivre's Theorem states that for any real number ? and integer n, the complex number (cos ? + i sin ?)^n is equal to cos(n?) + i sin(n?). This theorem simplifies the process of raising complex numbers expressed in polar form to a power and is essential in computations involving powers and roots of complex numbers.

Complex Conjugates

A complex conjugate is formed by changing the sign of the imaginary part of a complex number. If the number is a + bi, its conjugate is a - bi. Properties of complex conjugates are useful in simplifying expressions, particularly in operations like division, and they often appear in scenarios where symmetry in the complex plane plays a key role.

Complex Numbers

Complex numbers extend the real numbers by including the unit i, which is defined by the property i² = -1. They are written in the form a + bi, where a is the real part and b is the imaginary part. This concept provides a way to solve equations that have no real solutions and has applications in many areas of mathematics and engineering.

Polar Form Representation

The polar form of a complex number expresses it in terms of a magnitude and an angle relative to the positive real axis. A complex number z = a + bi can be written as r(cos ? + i sin ?) or r·e^(i?), where r is the modulus of z and ? is the argument. This representation is particularly useful for multiplication, division, and exponentiation of complex numbers.

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