Problem 28 - Precalculus: Functions and Graphs

Question

Write the expression in the form \(a+b i,\) where \(a\) and \(b\) are real numbers. $$(3-2 i)^{3}$$

Step-by-Step Solution

Answer

The expression in the form \(a + bi\) is \(-81 - 46i\).

Step 1: Identify the Expression Type

The expression \((3-2i)^3\) is a complex number raised to a power. To simplify it and express it in the form \(a+bi\), we need to expand the expression.

Step 2: Use Binomial Expansion

Use the binomial theorem for expansion: \((a-b)^3 = a^3 - 3a^2b + 3ab^2 - b^3\). Here, identify \(a = 3\) and \(b = 2i\), then plug these into the expansion.

Step 3: Expand each Term

Calculate each term separately: \(3^3 = 27\), \(-3 imes 3^2 imes 2i = -54i\), \(3 imes 3 imes (2i)^2 = 3 imes 9 imes (-4) = -108\), and \(-(2i)^3 = -8i^3 = -8(-i) = 8i\).

Step 4: Combine Like Terms

Add the real and imaginary parts separately. The real part: \(27 - 108 = -81\). The imaginary part: \(-54i + 8i = -46i\).

Step 5: Write the Final Form

Combine the results into the form \(a + bi\), giving us \(-81 - 46i\).

Key Concepts

Binomial Expansion

The binomial expansion is a method used to expand expressions that are raised to a power, specifically binomials which are expressions with two terms. In this exercise, the expression \((3-2i)^3\) needs to be rewritten into a simpler form. To do this, we use the Binomial Theorem, which provides a systematic way of expanding binomials. The theorem states:

• For any positive integer \(n\), the expansion of \((a + b)^n\) is given by: \[(a+b)^n = a^n + \binom{n}{1} a^{n-1}b + \binom{n}{2} a^{n-2}b^2 + \dots + b^n\]

In our exercise, we apply it to \((3-2i)^3\), where \(n=3\), \(a=3\), and \(b=-2i\). This leads to the following terms from the expansion:

• \(3^3 = 27\)
• \(-3 \times 3^2 \times 2i = -54i\)
• \(3 \times (2i)^2 = 3 \times -4 = -12\) (not \(-108\) which results from distributing \(-3\))
• \((2i)^3 = 8i\)

With this expansion, we break it down term by term, which simplifies the process of combining them into a single expression.

Imaginary Numbers

Imaginary numbers are a special category of numbers that emerge when we incorporate the square root of negative numbers. The way we deal with this is by using the imaginary unit, denoted as \(i\), where \(i = \sqrt{-1}\).

• This allows us to solve equations like \(x^2 + 1 = 0\)
• Since the square of any real number is positive, we use \(i\) to denote solutions of negative square roots.

In the context of the exercise, when calculating \((2i)^2\), we find \(-4\), representing the negative square, and \((2i)^3\), leads to \(-8i\), as it keeps the form intact but changes signs depending on the powers of \(i\).
Imaginary parts are combined separately from real parts in multistep calculations, exemplifying how they are best handled in arithmetic manipulation.

Real Numbers

Real numbers include all the familiar numbers you use in everyday life, such as integers, fractions, and decimals. They incorporate both rational numbers (which can be expressed as the quotient of two integers) and irrational numbers (which cannot be expressed as such fractions).
In the problem, we separately handle these real and imaginary components, identifying specific real terms within the expansion. For example, when simplifying \((3-2i)^3\), after all calculations, you isolate the real numbers from the imaginary part.

• For example, the expression \(27 - 108\) results in a real sum of \(-81\).

This distinction between real and imaginary terms enables accurate simplification and completion of expression by ensuring the correct application of binomial expansion results into the standard form \(a+bi\), where \(a\) and \(b\) represent real numbers.