Problem 35 - College Algebra Essentials

Question

In Exercises \(29-44\), perform the indicated operations and write the result in standard form. $$ (-3-\sqrt{-7})^{2} $$

Step-by-Step Solution

Answer

So, the square of \(-3-\sqrt{-7}\) in standard form is \(2 + 6i\sqrt{7}\).

Step 1: Express complex number in standard form

In this step, represent the complex number in the form a + bi. Here, \(-3-\sqrt{-7}\) can be written as \(-3 - i\sqrt{7}\) as \(\sqrt{-1} = i\).

Step 2: Apply the exponent

The exponentiation requires us to square the complex number. Using the identity \((a - b)^{2} = a^{2} - 2ab + b^{2}\), we can write the square of \(-3 - i\sqrt{7}\) as \( (-3)^{2} - 2(-3)i\sqrt{7} + (i\sqrt{7})^{2} \).

Step 3: Simplify the expression

The expression \((-3)^{2} - 2(-3)i\sqrt{7} + (i\sqrt{7})^{2}\) simplifies to \(9 + 6i\sqrt{7} - 7\). This equals \(2 + 6i\sqrt{7}\) after combining the real number terms.

Key Concepts

Exponentiation of Complex Numbers

Exponentiation in the realm of complex numbers can seem daunting at first, but with the right approach, it becomes an extension of familiar algebraic rules. When we raise a complex number to a power, we are scaling and rotating that number in the complex plane. A complex number in standard form is written as a + bi, where 'a' and 'b' are real numbers, and 'i' is the imaginary unit representing \( \sqrt{-1} \). For instance, to square the complex number \( -3 - i\sqrt{7} \), we can follow a process similar to expanding \(a - b)^2\) in real numbers.

Simplifying Complex Number Expressions

Simplifying complex number expressions involves combining like terms and applying basic arithmetic operations. The goal is to express the complex number in standard form, which is the simplest form for visualization and further calculations. For example, after exponentiating the complex number \( -3 - i\sqrt{7} \) as outlined in our exercise, we reach the expression \( 9 + 6i\sqrt{7} - 7 \). Simplifying requires combining the real parts \(9 - 7\) and keeping the imaginary part \(6i\sqrt{7}\) as is, resulting in \(2 + 6i\sqrt{7}\), a much cleaner and more useful form in complex number algebra.

Complex Numbers Algebra

Complex numbers algebra involves operations such as addition, subtraction, multiplication, division, exponentiation, and finding roots among complex numbers. Algebraic manipulation with complex numbers follows the same order of operations as with real numbers, but there's an added focus on the imaginary unit \(i\). A key aspect is to always express your final answer in standard form, \( a + bi\), for clarity. Whether you're squaring a complex number or adding two together, maintaining this form allows for easy interpretation and further manipulation. For example, when we squared \( -3 - i\sqrt{7} \) at the beginning of our exercise, continuing through finding the square and simplifying it, we maintained standard form, making each step clear and logical.