Problem 33 - College Algebra

Question

Perform the indicated operations and write the result in standard form. $$ (-2+\sqrt{-4})^{2} $$

Step-by-Step Solution

Answer

The answer for the problem \((-2+\sqrt{-4})^{2}\) is \(-8i\).

Step 1 - Express Square Root in Terms of i

The square root of -4 is \(2i\). This is because \(i\) is defined as the square root of -1, so \(\sqrt{-4} = \sqrt{4} * \sqrt{-1} = 2i\). Thus, the original expression, \(-2+\sqrt{-4}\), becomes \(-2 + 2i\).

Step 2 - Squaring the Complex Number

The square of \(-2 + 2i\), sealed by \((-2 + 2i)^{2}\), is computed by squaring the entire expression. The square of a binomial \(a+b\), is given by \(a^2 + 2ab + b^2\). Thus, the expression becomes \((-2)^2 + 2*(-2)*(2i) + (2i)^2\), which simplifies to \(4 - 8i - 4\).

Step 3 - Simplify the Result

The calculation in the last step yields the number \(4 - 8i - 4\), which needs to be simplified to put in standard form \(a+bi\). After simplification, the result is \(-8i\)

Key Concepts

Standard Form

Understanding the standard form of a complex number is crucial for working with complex algebra. The standard form is written as \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit. In this form, \(a\) represents the real part of the complex number, and \(b\) represents the imaginary part. When we simplify expressions with complex numbers, we aim to express the final result as this standard form. This makes the complex numbers easier to interpret and use in further calculations.

For example, when simplifying the expression \((-2 + 2i)^2\), our goal is to represent the result in the form of \(a + bi\), where we identify and separate the real and imaginary parts. This aids in clarity and provides a structure that is readily usable for addition, subtraction, or comparing to other complex numbers.

Imaginary Unit

The concept of the imaginary unit is at the heart of complex numbers. The imaginary unit is denoted by \(i\) and is defined as the square root of \(-1\). It is important to note that \(i\) itself does not have a value that can be represented on the real number line. Instead, it exists as a mathematical construct that allows for the extension of the number system to include numbers whose squares are negative.

Returning to the exercise, the square root of \(-4\) is expressed in terms of \(i\) as \(2i\), because \(i = \sqrt{-1}\) and \(\sqrt{-4} = \sqrt{4} \times \sqrt{-1} = 2 \times i\). By understanding this, we can transform what may initially appear as an alarming square root of a negative number into a manageable expression involving \(i\).

Squaring Complex Numbers

Squaring complex numbers involves a process similar to squaring binomials in real-number algebra. Given a complex number in the form \(a + bi\), where \(a\) and \(b\) are real numbers and \(i\) is the imaginary unit, we square it by multiplying the number with itself. This is done using the formula \((a + bi)^2 = a^2 + 2abi + b^2i^2\).

In the given exercise, squaring the complex number \(-2 + 2i\) involves applying this formula. We get \((-2 + 2i)^2 = (-2)^2 + 2(-2)(2i) + (2i)^2\). When we simplify this, it is crucial to remember that \(i^2 = -1\), which will affect the calculation of the constant and the coefficient of \(i\). After carrying out the multiplication and combining like terms, we arrive at the final answer in standard form.