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

Verified step by step guidance

1. Step 1: Recall the standard form of a complex number, which is written as a + bi, where 'a' is the real part and 'b' is the imaginary part.
2. Step 2: Expand the expression (1 + i)^3 using the binomial theorem or repeated multiplication. The binomial theorem states (a + b)^n = Σ[k=0 to n] (n choose k) * a^(n-k) * b^k.
3. Step 3: Apply the binomial theorem to (1 + i)^3. Substitute a = 1, b = i, and n = 3. Compute each term of the expansion: (3 choose 0)(1^3)(i^0), (3 choose 1)(1^2)(i^1), (3 choose 2)(1^1)(i^2), and (3 choose 3)(1^0)(i^3).
4. Step 4: Simplify each term using the properties of exponents and the fact that i^2 = -1 and i^3 = -i. Combine like terms (real and imaginary parts).
5. Step 5: Write the final result in standard form, a + bi, by combining the real and imaginary components obtained from the expansion.

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.

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, this technique is useful for simplifying expressions like (1 + i)^3 before converting them to standard form.