Write each complex number in standard form. (1 - i)³

Verified step by step guidance

1. Rewrite the expression \((1 - i)^3\) using the binomial theorem, which states \((a + b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k\). Here, \(a = 1\), \(b = -i\), and \(n = 3\).
2. Expand the binomial expression: \((1 - i)^3 = \binom{3}{0}(1)^3(-i)^0 + \binom{3}{1}(1)^2(-i)^1 + \binom{3}{2}(1)^1(-i)^2 + \binom{3}{3}(1)^0(-i)^3\).
3. Simplify each term: \(\binom{3}{0}(1)^3(-i)^0 = 1\), \(\binom{3}{1}(1)^2(-i)^1 = -3i\), \(\binom{3}{2}(1)^1(-i)^2 = 3(-1) = -3\), and \(\binom{3}{3}(1)^0(-i)^3 = -i\).
4. Combine the simplified terms: \(1 - 3i - 3 - i\).
5. Group the real and imaginary parts to write the complex number in standard form \(a + bi\), where \(a\) is the real part and \(b\) is the coefficient of \(i\).

Key Concepts

Complex Numbers

Complex numbers are numbers that have a real part and an imaginary part, expressed in the form a + bi, where a is the real part, b is the imaginary part, and i is the imaginary unit defined as the square root of -1. Understanding complex numbers is essential for manipulating and performing operations on them, such as addition, subtraction, multiplication, and exponentiation.

Standard Form of Complex Numbers

The standard form of a complex number is a + bi, where a and b are real numbers. To express a complex number in standard form, one must ensure that the imaginary unit i is isolated in the second term. This is crucial when simplifying expressions involving complex numbers, as it allows for easier interpretation and further calculations.

Exponentiation of Complex Numbers

Exponentiation of complex numbers involves raising a complex number to a power, which can be done using the binomial theorem or De Moivre's theorem. In the case of (1 - i)^3, one can expand the expression using the binomial theorem, which states that (a + b)^n can be expressed as a sum of terms involving combinations of a and b raised to various powers. This process is key to simplifying complex expressions.