Problem 22 - Advanced Engineering Mathematics

Question

In Problems 1-26, write the given number in the form \(a+i b\). $$ (1+i)^{2}(1-i)^{3} $$

Step-by-Step Solution

Answer

The answer is \(4 - 4i\).

Step 1: Expand Using Binomial Theorem

First, expand \[ (1 + i)^2 = (1 + i)(1 + i) = 1 + i + i + i^2 = 1 + 2i + i^2 \]Since \(i^2 = -1\), this becomes:\[ 1 + 2i - 1 \] Simplifying gives:\[ 2i \]

Step 2: Expand Using Binomial Theorem for Cube

Next, expand \[ (1 - i)^3 = (1-i)(1-i)(1-i) \] First expand the first two terms:\[ (1-i)(1-i) = 1 - i - i + i^2 = 1 - 2i + i^2 \]Since \(i^2 = -1\), it becomes:\[ 1 - 2i - 1 = -2i \]Next, multiply by the remaining \(1-i\):\[ (-2i)(1-i) = -2i + 2i^2 \]Replacing \(i^2\) with \(-1\):\[ -2i - 2 = -2 - 2i \]

Step 3: Multiply the Results

Now multiply the results from steps 1 and 2:\[ (2i)(-2 - 2i) \]Using distributive property:\[ 2i(-2) + 2i(-2i) \]\[ = -4i - 4i^2 \]Since \(i^2 = -1\), it becomes:\[ -4i + 4 \]Thus, in terms of \(a + ib\), it is:\[ 4 - 4i \]

Step 4: Write the Result in Standard Form

The final computation results in the number format \(4 - 4i\), which is already in the form \(a + ib\) where \(a = 4\) and \(b = -4\).

Key Concepts

Binomial Theorem

The Binomial Theorem is an essential concept in algebra that provides a quick way to expand expressions that are raised to a power. When we have an expression like \((1 + i)^2\), the theorem helps us avoid tedious multiplication by following a specific pattern of expansion. This powerful tool allows us to break down expressions such as \((1+i)^n\) into simpler, more manageable terms. Using the theorem, we apply coefficients from binomial expansions called binomial coefficients.

• For \((1 + i)^2\), it essentially involves multiplying \((1 + i)(1 + i)\).
• It leads to terms \(1 + 2i + i^2\), which further simplify when you recognize \(i^2 = -1\).

This simplification results in \(1 + 2i - 1 = 2i\). The Binomial Theorem simplifies the process of expanding complex powers.

Imaginary Unit

The imaginary unit, represented by \(i\), is a fundamental concept in complex numbers. It is the building block that allows us to extend the real number system to include complex numbers. The defining property of \(i\) is that it's the square root of \(-1\):

• \(i^2 = -1\) is the cornerstone property.
• It helps simplify expressions involving squares, leading to real number results in combination with imaginary parts.

In our problem, you can see how \(i^2 = -1\) plays a crucial role in the step-by-step expansion and simplification of expressions like \((1 + i)^2\) and \((1-i)^3\). Understanding \(i\) is essential for working with any complex number, as it forms the basis for separating real and imaginary parts in expressions.

Algebraic Simplification

Algebraic simplification is the process of breaking down complex algebraic expressions into simpler forms. When dealing with complex numbers, this not only involves manipulating numeric coefficients but also properly managing the imaginary unit \(i\). Simplification involves:

• Identifying and combining like terms.
• First expanding the expression using distributive properties.
• Substituting known values like \(i^2 = -1\) to further simplify.

In the solution, after expanding \((1 + i)^2\) and \((1 - i)^3\), correct simplification transforms them into \(2i\) and \(-2 - 2i\) respectively. Finally, multiplication and additional simplification steps provide the result \(4 - 4i\). The process emphasizes the combination of both algebraic manipulation and the properties of the imaginary unit.