Problem 35 - College Algebra

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

The result of squaring (-3 - sqrt(-7)) is (2 + 6i*sqrt(7)).

Step 1: Rewrite sqrt(-7)

Rewrite sqrt(-7) as i * sqrt(7). So, we have: (-3 - i*sqrt(7))^{2}.

Step 2: Expand the square

Using the formula (a-b)^{2}=a^{2}-2ab+b^{2}, we expand the square: (-3^{2} - 2(-3)(i*sqrt{7}) + (i*sqrt{7})^{2}). Simplifying, we get: (9 + 6i*sqrt{7} - 7).

Step 3: Combine real parts

The expression (9 + 6i*sqrt{7} - 7) has two real parts: 9 and -7. Combine these to get: (2 + 6i*sqrt(7)).

Key Concepts

Imaginary Unit

Complex numbers are built upon the foundation of real numbers by introducing the concept of the imaginary unit. The imaginary unit is symbolized by the letter \(i\) and is defined by the equation \(i^2 = -1\). This definition arises from the need to express solutions to equations that do not have real solutions, such as the square root of negative numbers.

In practical terms, when dealing with the square root of a negative number, we can rewrite it with \(i\). For example, if you have \(\sqrt{-7}\), it can be rewritten as \(i\sqrt{7}\). This transformation from a real impossible square root to an imaginary number allows us to perform further mathematical operations with complex numbers that include both real and imaginary parts.

Square Root

The square root operation is a basic yet crucial mathematical function that can sometimes lead to the introduction of complex numbers. It involves finding a number which, when multiplied by itself, gives the original number. Square roots of positive numbers are well-defined in the real number system; however, when it comes to negative numbers, we invoke the imaginary unit \(i\).

For instance, consider the square root of \(-7\). Since there is no real number whose square is \(-7\), we express \(\sqrt{-7}\) as \(i\sqrt{7}\) using the imaginary unit.
This conversion is a key step in complex number arithmetic, allowing expressions involving negative square roots to be manipulated and simplified further.

Binomial Expansion

Binomial expansion is a method used to expand expressions raised to a power, and it's particularly handy when working with complex numbers. The formula \((a - b)^2 = a^2 - 2ab + b^2\) is known as a special case of the binomial theorem. It allows for the efficient expansion of the squared binomial expressions.

For example, in the expression \((-3 - i\sqrt{7})^2\), it acts like this:

• First term: \((-3)^2 = 9\)
• Second term: \(-2 \times (-3) \times (i\sqrt{7}) = 6i\sqrt{7}\)
• Third term: \((i\sqrt{7})^2 = i^2 \times 7 = -7\)

Adding these expanded parts together, the result combines the real terms (9 and -7) and the imaginary term \(6i\sqrt{7}\), effectively transforming it into a simplified complex number expression.

Standard Form

When dealing with complex numbers, it's essential to express them in a consistent and clear format known as the "standard form." The standard form of a complex number is \(a + bi\), where \(a\) is the real part, and \(bi\) is the imaginary part.

In this format, \(a\) and \(b\) are real numbers. The "\(i\)" indicates the presence of the imaginary part. This notation not only standardizes the way complex numbers are written but also makes it easier to perform operations such as addition, subtraction, and multiplication.

• For example, after expanding \((-3 - i\sqrt{7})^2\), we obtain \(2 + 6i\sqrt{7}\).

In this expression, \(2\) is the real part, and \(6i\sqrt{7}\) is the imaginary part, perfectly fitting the standard form \(a + bi\). Standard form makes complex arithmetic more accessible and intuitive.