Question

Justify each step in performing the operation. $$ \begin{aligned} (3&+2 i)(7-4 i) \\ &=21-12 i+14 i-8 i^2 \\ &=21+2 i-8(-1) \\ &=21+2 i+8 \\ &=29+2 i \end{aligned} $$

Short answer

The step-by-step justification for the given complex multiplication operation indicates that the process has been performed correctly, giving the final result as \(29 + 2i\).

Step-by-step solution

Distribute multiplications

Distribute the multiplication across both terms in the second complex number, according to the distributive property of multiplication over addition, i.e., \(a(b + c) = ab + ac\). Therefore, \((3+2i)(7-4i) = 3*(7-4i) + 2i*(7-4i) = 21-12i + 14i - 8i^2\).

Simplify terms

Combine like terms, in this case, the imaginary components: \(-12i + 14i = 2i\). Thus the expression simplifies further to: \(21 + 2i - 8i^2\).

Replace \(i^2\)

In the case of complex numbers, replace \(i^2\) by its definition. Recall that \(i^2 = -1\), as the definition of imaginary numbers. This results in: \(21 + 2i - 8*(-1)\).

Final simplification

Do the multiplication and addition operations to obtain the final result. The multiplication operation gives us: \(21 + 2i + 8\), which simplifies to: \(29 + 2i\).

Key concepts

Distributive Property

The distributive property is a fundamental rule in algebra that helps in multiplying a number across terms inside parentheses. It says that multiplying a single number by two or more terms inside a bracket is the same as multiplying each term individually and then adding them together. In other words, for any numbers \(a\), \(b\), and \(c\), the distributive property ensures that \(a(b + c) = ab + ac\).

When it comes to complex numbers, the distributive property works the same way. Consider

• \((3 + 2i)(7 - 4i)\):

Here, you distribute each term from \((3 + 2i)\) to both terms of \((7 - 4i)\).

• First, multiply \(3\) by both \(7\) and \(-4i\), which gives \(21 - 12i\).
• Then, multiply \(2i\) by both \(7\) and \(-4i\), which results in \(14i - 8i^2\).

You now have all the terms required in expanding the expression.

Imaginary Unit

The imaginary unit 'i' is a central concept in complex numbers. It is defined as the square root of \(-1\), i.e., \(i = \sqrt{-1}\).

When working with complex numbers, we often encounter \(i^2\). Understanding \(i^2\) is crucial because

• \(i^2\) directly equals \(-1\).

In the problem at hand,

• note how \(i^2 = -1\) and is used to transform \(-8i^2\) into \(8\).

These transformations allow operations involving complex numbers to be simplified, enabling universally applicable arithmetic processes on them.

Combining Like Terms

Combining like terms is a method used to simplify expressions in algebra that have common elements. This process involves adding or subtracting coefficients of the terms that have the same variables.

In the context of complex numbers, like terms refer to terms involving similar powers of the imaginary unit \(i\). For instance,

• If you have \(-12i + 14i\), you can combine them because they both are coefficients of the same imaginary unit \(i\).

Subtracting \(12i\) from \(14i\) gives \(2i\).

This step simplifies the expression significantly and is crucial in reaching the final simplified form of complex numbers.

Simplifying Expressions

Simplifying expressions is all about reducing an equation or formula to its simplest form. This involves performing operations like addition, subtraction, multiplication, division, and combining like terms.

After applying the distributive property and combining like terms, the next step is to replace \(i^2\) with \(-1\) and simplify further. In the given exercise:

• The expression becomes \(21 + 2i - 8i^2\), which simplifies to \(21 + 2i + 8\).

Then, by further combining constant terms:

• \(21 + 8\) results in \(29\).

Thus, the simplified result of the expression is \(29 + 2i\). Simplifying expressions is essential for clear and concise calculations, making it easier to work with and solve complex mathematical problems.