Problem 28 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-34: 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

(-9 - 46i)

Step 1: Expand the Expression Using Binomial Theorem

The expression \((3 - 2i)^3\) is a binomial raised to a power. Use the Binomial Theorem, which states \((x+y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k\) to expand. In our case, \(x=3\), \(y = -2i\), and \(n=3\). The expanded expression is: \[(3)^3 + 3\cdot(3)^2\cdot(-2i) + 3\cdot(3)\cdot(-2i)^2 + (-2i)^3\]

Step 2: Calculate Each Term Separately

Calculate each term of the expansion: - \((3)^3 = 27\) - \(3 \cdot (3)^2 \cdot (-2i) = 3 \cdot 9 \cdot (-2i) = -54i\) - \(3 \cdot 3 \cdot (-2i)^2 = 9 \cdot (-4) = -36\) (note \((-2i)^2=(-2)^2(i)^2 = 4(-1) = -4\)) - \((-2i)^3 = -8i\cdot i^2 = -8i(-1) = 8i\) (note \(i^3 = i\cdot i^2 = -i\))

Step 3: Combine Like Terms

Combine the real and imaginary terms from the expanded form: - Real terms: \(27 - 36 = -9\) - Imaginary terms: \(-54i + 8i = -46i\) Thus, the expression simplifies to \(-9 - 46i\).

Step 4: Final Expression in Standard Form

Combine the computed real and imaginary parts to write the final result in the form \(a + bi\): - The expression \((3 - 2i)^3\) is \(-9 - 46i\).

Key Concepts

Binomial Theorem

The Binomial Theorem is a powerful tool that helps us expand expressions involving binomials raised to powers. A binomial is simply an expression that contains two terms, for example,

• In your textbook exercise, the formula for the Binomial Theorem looks like this: \((x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k\).

Here, \(x\) and \(y\) are the terms of the binomial, \(n\) is the exponent, and \(\binom{n}{k}\) represents the binomial coefficient. This coefficient selects different combinations of \(x\) and \(y\) as they appear in the expansion.
For your task, the relevant binomial is \((3 - 2i)^3\). Here, your values are \(x = 3\), \(y = -2i\), and \(n = 3\). By applying the theorem, you expand the expression into separate terms that are easier to handle one by one.

Imaginary Numbers

Imaginary numbers might sound exotic, but they have practical uses in fields like engineering and physics. The basic building block of imaginary numbers is \(i\), which is defined as \(i^2 = -1\).
This means sequences like \((i^2 = -1)\), \( (i^3 = i \, \text{•}\, i^2 = -i)\), and launching into repeated cycles \((i^4=1)\), and so forth.

• When working through the binomial expansion: - The \((-2i)^2\) becomes \((-2)^2\cdot(i)^2 = 4 \cdot (-1) = -4\). - Similarly, \((-2i)^3\) takes it further, as \(= -8i\), then \(i^3\) becomes \(-i\), resulting in \(-8i \, \cdot\, (-1) = 8i\).

Understanding how \(i\) operates in calculations is key to solving expressions like this one.

Algebraic Expansion

Algebraic expansion is the process of transforming a product of sums into multiple simpler terms. In the exercise, the hard work needed to manage \((3 - 2i)^3\) is simplified by expansion.

• First, recognize the individual elements generated using the Binomial Theorem.

• Then, compute each term: -\((3)^3\) becomes straightforward as \(27\).
• The next element, \(3\,\cdot\,(3)^2\cdot(-2i) \), converts into \(-54i\).
• Next, \(3 \cdot 3 \cdot (-2i)^2 = -36\).
• Finally, \(-2i)^3\), simplifies to \(8i\).

• The last key step is combining like terms: bringing together real numbers and imaginary numbers here to eventually get \(-9 - 46i\).

Breaking down a complicated expression like this involves mixing binomial theory with familiarity with imaginary numbers—and a keen eye for detail!