Problem 34 - College Algebra

Question

In Exercises \(29-44,\) perform the indicated operations and write the result in standard form. $$(-5-\sqrt{-9})^{2}$$

Step-by-Step Solution

Answer

The result of \((-5-\sqrt{-9})^{2}\) in standard form is 64.

Step 1: Identify the form

Identify the given expression as a complex number in the form \(a-bi\), where \(a = -5\) and \(b = \sqrt{9}\) . The value of \(i\) is defined as \(i^2 = -1\).

Step 2: Apply binomial square formula

Apply the formula for \((a-b)^2\) which is \(a^2 - 2ab + b^2\). Here \(a = -5\) and \(b = \sqrt{9} = 3\) . Substituting these values results in \((-5)^2 - 2*(-5)*3 + (3i)^2\).

Step 3: Calculate the square

Compute each term independently, giving \(25 + 30 - 9i^2\).

Step 4: Substitute \(i^2 = -1\)

Substitute \(i^2\) with its value \(-1\) to get \(25 + 30 - 9*-1\).

Step 5: Simplify.

After simplifying \(25 + 30 - 9*-1\), we get 64, which is the result in standard form.

Key Concepts

Imaginary Unit

An imaginary unit is a fundamental concept in mathematics, especially when dealing with complex numbers. Imaginary numbers are essential in solving certain algebraic equations where real numbers fall short. The most basic imaginary unit is represented as \(i\). This unit is defined by the essential property:

• \(i^2 = -1\)

This definition allows us to expand the real number system to include complex numbers. Complex numbers are of the form \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part.
The imaginary unit makes it possible to perform operations on numbers that were thought to have no real solutions. For instance, the square root of a negative number, like \(\sqrt{-9}\), cannot be solved with real numbers alone. Instead, we rewrite it as \(3i\), because \(\sqrt{-9} = \sqrt{9} \times i = 3i\).
Thus, understanding the imaginary unit \(i\) is crucial for manipulating and solving equations within the realm of complex numbers.

Complex Conjugate

A complex conjugate is a related concept that helps simplify the operations with complex numbers. If you have a complex number \(a + bi\), its complex conjugate is \(a - bi\). Essentially, you flip the sign of the imaginary component.

• Original: \(a + bi\)
• Conjugate: \(a - bi\)

The complex conjugate is incredibly useful for simplifying division of complex numbers or finding magnitudes. When you multiply a complex number by its conjugate, the result is a real number, a property rooted in the imaginary unit \(i^2 = -1\).
For example, consider the complex number \(-5 - 3i\). Its complex conjugate is \(-5 + 3i\). Multiplying these gives \[(-5 - 3i)(-5 + 3i) = (-5)^2 - (3i)^2=25-9(-1)=25+9=34\] Here, thus, verifying the operation results in a real number. Working with complex conjugates is a key part of handling complex number computations effectively.

Binomial Theorem

The binomial theorem is a powerful tool in mathematics that provides a formula to expand expressions raised to a power. It is not only limited to real numbers but also extends to complex numbers. The specific expression for dealing with squares, like in our exercise, is known as the binomial square formula. For a binomial expression \((a-b)^2\), the formula is:\[(a-b)^2 = a^2 - 2ab + b^2\]This formula allows us to break down and simplify expressions by working out each part separately.
In the context of complex numbers, such as the expression \((-5 - \sqrt{-9})^2\), this formula helps in simplifying: 1. Use \(a = -5\) and \(b = 3i\) to apply the formula as \((-5)^2 - 2(-5)(3i) + (3i)^2\).2. Calculate each term separately to get \(25 + 30i - 9i^2\).3. Substitute \(i^2 = -1\) into the expression to simplify to a standard form.The binomial theorem, and particularly its square form, is a foundational concept for addressing complex numbers, allowing us to break down and compute otherwise challenging expressions with relative ease.