Problem 33 - College Algebra Essentials

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

The operation \((-2+\sqrt{-4})^{2}\) simplifies to \(-8i\). Therefore, the result in the standard form of the operation is \(-8i\).

Step 1: Simplify the square root of the negative number

When a negative number is under a square root, it's outcome is an imaginary number. So, \(\sqrt{-4}\) can be written in terms of 'i', where \(i^2 = -1\), so \(\sqrt{-4}\) can be simplified as \(2i\). This makes the expression inside parentheses as \(-2 + 2i\)

Step 2: Apply the square of binomial formula

Now apply the formula \((a+b)^2 = a^2 + 2ab + b^2\). Here, a is -2 and b is 2i. So \((-2 + 2i)^2 = (-2)^2 + 2*(-2)*(2i) + (2i)^2 = 4 - 8i -4\)

Step 3: Simplify the expression

After the operations, simplify the expression to find the final answer. The final expression is \(4 - 8i - 4 = -8i\)

Key Concepts

Imaginary Unit

The imaginary unit, denoted as \(i\), is a fundamental concept when dealing with complex numbers. It stems from the necessity of finding square roots of negative numbers. The defining property of \(i\) is that \(i^2 = -1\). This simple yet powerful property allows mathematicians to extend the scope of real numbers and solve equations that are otherwise impossible with only real numbers.

Here's why \(i\) is so useful:

• It creates a new dimension for numbers, allowing us to solve equations that include the square root of negative numbers.
• It combines with real numbers to form complex numbers, which are written as \(a + bi\), where \(a\) and \(b\) are real numbers.
• It makes it possible to represent quantities that cannot be expressed as real numbers alone, such as impedance in electrical engineering.

Understanding the imaginary unit is the first step to grasping complex numbers and their application in various fields.

Binomial Theorem

The Binomial Theorem is a crucial tool in algebra that provides a way to expand expressions of the form \((a+b)^n\). In the context of complex numbers, it helps us to expand expressions involving imaginary numbers. It states that:\[(a+b)^2 = a^2 + 2ab + b^2\]

Using the Binomial Theorem helps simplify computations involving sums of a real number and an imaginary number. For example, if you have the expression \((-2 + 2i)^2\), here's what happens:

• Square the real part: \((-2)^2 = 4\).
• Double the product of the real and imaginary parts: \(2\times (-2) \times 2i = -8i\).
• Square the imaginary part: \((2i)^2 = 4i^2 = -4\) since \(i^2 = -1\).

Combine all these to get the expanded form of the expression.

Complex Conjugate

The complex conjugate is a key concept in handling complex numbers, particularly useful when simplifying expressions. If you have a complex number \(z = a + bi\), its complex conjugate is \(\overline{z} = a - bi\).

The properties and uses of the complex conjugate are:

• When a complex number is multiplied by its conjugate, the result is a real number: \((a + bi)(a - bi) = a^2 + b^2\).
• This feature is especially useful in division and simplifying expressions, making it easier to find real results from complex operations.
• In geometry, it reflects the number across the real axis in the complex plane, giving useful symmetrical properties.

Conjugates make handling complex numbers simpler and are instrumental in many mathematical operations.

Square Root of Negative Numbers

Taking the square root of a negative number is impossible in the world of real numbers, which is where imaginary numbers come in. When you see a negative under the square root, like \(\sqrt{-4}\), it's a call to use the imaginary unit \(i\). This is because:\\(sqrt{-4} = \sqrt{4} \cdot \sqrt{-1} = 2i\)\

Why this matters:

• Provides a way to work with numbers outside the real number system, expanding mathematical possibilities.
• Transforms equations and expressions involving negative roots into workable forms using \(i\).
• Underpins the entire system of complex numbers, which encompasses both real and imaginary parts.

Understanding how to handle the square root of negative numbers is crucial for working seamlessly with complex numbers.