In Exercises 61–64, write each complex number in standard form. (1 + i)^3

Verified step by step guidance

1. Start by recognizing that the expression \((1 + i)^3\) involves raising a binomial to a power. Use the binomial theorem or expand it directly.
2. Apply the binomial expansion: \((a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3\). Here, \(a = 1\) and \(b = i\).
3. Calculate each term: \(1^3 = 1\), \(3 \times 1^2 \times i = 3i\), \(3 \times 1 \times i^2 = 3i^2\), and \(i^3 = i \times i^2 = i \times (-1) = -i\).
4. Substitute these values back into the expansion: \(1 + 3i + 3i^2 - i\).
5. Remember that \(i^2 = -1\), so replace \(3i^2\) with \(3(-1) = -3\). Combine like terms to write the complex number in standard form \(a + bi\).

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 form is crucial for clarity and consistency in mathematical communication, especially when performing operations or comparisons between complex numbers.

Binomial Expansion

Binomial expansion is a method used to expand expressions that are raised to a power, such as (a + b)^n. The expansion can be achieved using the Binomial Theorem, which provides a formula for calculating the coefficients of the terms in the expansion. In the context of complex numbers, binomial expansion is particularly useful for simplifying expressions like (1 + i)^3, allowing for the calculation of powers of complex numbers.