Problem 33 - College Algebra

Question

In Exercises \(29-44,\) perform the indicated operations and write the result in standard form. $$(-2+\sqrt{-4})^{2}$$

Step-by-Step Solution

Answer

\(- 8i\)

Step 1: Simplify the square root

Start by simplifying \(\sqrt{-4}\). Remember that the square root of -1 is \(i\), and that the square root of 4 is 2. So, \(\sqrt{-4}\) equals \(2i\). Replace \(\sqrt{-4}\) with \(2i\) in the original expression to get \(-2 + 2i\).

Step 2: Perform the square operation

Now, perform the square operation on \(-2 + 2i\). Remember that squaring a binomial means multiplying it by itself. So we have \((-2 + 2i) * (-2 + 2i)\). Distribute each term in the first binomial to each term in the second to get \(4 - 4i - 4i + 4i^{2}\).

Step 3: Simplify the expression

To simplify, remember that \(i^{2} = -1\). Replace \(i^{2}\) with \(-1\), to get \(4 - 4i - 4i - 4\). Combine like terms to get \(- 8i\). This is the result in standard form.

Key Concepts

Standard Form of a Complex Number

Understanding the standard form of a complex number is crucial for performing arithmetic operations with complex numbers. A complex number consists of a real part and an imaginary part, and the standard form is written as \(a + bi\), where \(a\) is the real component and \(b\) represents the imaginary component, with \(i\) being the imaginary unit.

For instance, in the given exercise, we simplify \(\sqrt{-4}\) to \(2i\) and thus the complex number \( -2 + \sqrt{-4} \) becomes \( -2 + 2i\) in standard form. It's crucial to get comfortable with converting complex expressions into this form, as it makes subsequent operations like addition, subtraction, multiplication, or even finding the complex conjugate much more straightforward.

Simplifying Square Roots

Simplifying square roots, especially those involving negative numbers, is a common task in algebra. The key to simplifying a square root of a negative number is to recognize that the square root of \( -1 \) is defined as the imaginary unit \( i \.\)

Therefore, any square root of a negative number can be simplified by finding the square root of its positive counterpart and then multiplying that by \(i\). For example, in the original exercise, \( \sqrt{-4} \) is simplified by taking the square root of the positive number \(4\), which is \(2\), and then multiplying by \(i\) to get \(2i\). This step is the foundation for working with complex numbers.

Binomial Multiplication

Binomial multiplication is a fundamental concept in algebra that involves multiplying two binomials together. To multiply binomials, we use the distributive property also known as FOIL (First, Outer, Inner, Last) which ensures each term in the first binomial is multiplied by each term in the second.

In the provided exercise, the binomial \( -2 + 2i \) is squared, meaning it is multiplied by itself. When expanded, this results in four terms: \( (-2)\times(-2), (-2)\times(2i), (2i)\times(-2), (2i)\times(2i)\), which simplifies to \(4 - 4i - 4i + 4i^2\). Understanding this process allows students to simplify expressions involving binomials and complex numbers.

Imaginary Unit \(i\)

The imaginary unit, designated as \(i\), is a mathematical concept used to represent the square root of \( -1 \.\) It's an essential part of complex numbers and it obeys certain algebraic rules, the most important being that \( i^2 = -1 \.\)

In the context of our exercise, once we've multiplied our binomials to get terms like \(4i^2\), we can replace \(i^2\) with \( -1 \) to further simplify our expression. Recognizing and applying the properties of the imaginary unit is key in performing operations involving complex numbers and is a building block for more advanced topics in mathematics including calculus and differential equations.