Problem 28 - Algebra and Trigonometry with Analytic Geometry

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 \(-9 - 62i\).

Step 1: Expand the expression

We need to expand the cube of the complex number \((3 - 2i)^3\). Apply the binomial theorem for expansion: \((a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3\). Substitute \(a = 3\) and \(b = 2i\), giving: \[ (3 - 2i)^3 = (3)^3 - 3(3)^2(2i) + 3(3)(2i)^2 - (2i)^3. \] Expand this step by step.

Step 2: Calculate each term

- Calculate \( (3)^3 = 27 \).- Calculate \( -3(3)^2(2i) = -3 \times 9 \times 2i = -54i \).- Calculate \( 3(3)(2i)^2 = 3 \times 3 \times (-4) = -36 \) (note that \( (2i)^2 = 4i^2 = -4 \)).- Calculate \( -(2i)^3 = -(8i) \) (note that \( (2i)^3 = 8i^3 = -8i \) because \(i^2 = -1\)). Now, substitute these results back into the equation.

Step 3: Combine the terms

Combine all the calculated terms: \[ 27 - 54i - 36 - 8i = (27 - 36) + (-54i - 8i) = -9 - 62i. \] Thus, the expression \((3 - 2i)^3\) simplifies to \(-9 - 62i\).

Step 4: Express in the form \(a + bi\)

The expression \((3 - 2i)^3 = -9 - 62i\) is already in the form \(a + bi\), where \(a = -9\) and \(b = -62\).

Key Concepts

Binomial Theorem

The Binomial Theorem is a powerful tool that allows us to expand expressions raised to a power, such as \((a + b)^n\). In the context of complex numbers, it becomes very useful for simplifying expressions like \((3 - 2i)^3\), as it provides a systematic way to break down the expansion.When you use the Binomial Theorem for an expression like \((a - b)^3\), the formula you apply is:

• \(a^3 - 3a^2b + 3ab^2 - b^3\)

Here, you simply replace \(a\) and \(b\) with the terms from your expression. This breaks the problem into smaller, more manageable pieces. If applying it to \((3 - 2i)^3\), substitution gives us:

• \((3)^3 - 3(3)^2(2i) + 3(3)(2i)^2 - (2i)^3\)

After expanding, you'll combine like terms to get the final result. The Binomial Theorem is fundamental not only in algebra but also in understanding polynomial expansions in complex numbers.

Imaginary Unit

The imaginary unit, often denoted as \(i\), is a core concept in dealing with complex numbers. It is defined as:

• \(i = \sqrt{-1}\)

This definition implies that when squared, \(i\) results in \(i^2 = -1\). Using this property becomes essential as you handle equations involving powers of \(i\).In our problem, when calculating terms like \((2i)^2\), first square the coefficient \(2\) to get \(4\), then multiply by \(i^2\) (which is \(-1\)). This gives \( (2i)^2 = 4i^2 = -4 \). Similarly, \((2i)^3 = 8i^3\) simplifies using \(i^3 = i \times i^2 = i(-1) = -i\), resulting in \(-8i\).Understanding the imaginary unit helps us tackle the complex nature of these calculations by turning seemingly unsolvable square roots and powers into manageable parts that contribute to the solution of complex equations.

Complex Expansion

Expanding expressions that involve complex numbers, such as \((3 - 2i)^3\), brings together several pivotal concepts in algebra and complex number theory. The goal of complex expansion is to simplify so we can express these terms in the form \(a + bi\), where \(a\) and \(b\) are real numbers. This often involves:

• Applying algebraic identities, like the Binomial Theorem
• Understanding properly how to manage powers of \(i\)
• Carefully combining like terms

For example, the expansion \((3 - 2i)^3\) uses the Binomial Theorem to find individual components step by step, such as \(27\), \(-54i\), \(-36\), and \(-8i\). Finally, combining these gives \(-9 - 62i\), clearly showing real and imaginary parts.With complex expansion, the goal is a neat and organized expression that separates real part \(a\) from the imaginary part \(bi\). This clarity is vital as it translates to real-world applications of complex numbers, including engineering and physics.