Problem 18 - College Algebra Essentials

Question

In Exercises \(9-20,\) find each product and write the result in standard form. $$(-7-i)(-7+i)$$

Step-by-Step Solution

Answer

The product of the two complex numbers (-7-i) and (-7+i) is 48 in the standard form.

Step 1: Distribute the multiplication

To solve this, the first step is to distribute the multiplication inside the brackets similar to how you would solve it in the case of normal brackets: \[(-7 - i)(-7 + i) = (-7)*(-7) + (-7)*i + (-i)*(-7) + (-i)*i.\]

Step 2: Solve the multiplication

Now, solve each multiplication separately: \[(-7)*(-7) = 49, (-7)*i = -7i, (-i)*(-7) = 7i,(-i)*i = -1. \] So the equation is now: \[= 49 - 7i + 7i -1.\]

Step 3: Simplify

Now simplify the equation by adding the like terms. Here, the -7i and 7i will cancel out each other: \[= 49 - 1 = 48.\]

Key Concepts

Multiplication of Complex Numbers

When multiplying complex numbers, it's important to remember that a complex number is usually expressed in the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit, \(i^2 = -1\). The multiplication of two complex numbers involves distributing each part of the complex numbers, much like polynomial multiplication.

For example, given two complex numbers, \((-7 - i)\) and \((-7 + i)\), you distribute each element as follows:

• Multiply the real parts: \((-7) \cdot (-7)\),
• Multiply the outer terms: \((-7) \cdot i\),
• Multiply the inner terms: \(-i \cdot (-7)\),
• Multiply the imaginary parts: \(-i \cdot i\).

Each operation is straightforward, and you need to take into account the defining characteristic of the imaginary unit: \(i^2 = -1\). Once the distributive multiplication is complete, you add the resulting products together. The final step is simplifying the expression and combining like terms to streamline it into a result, often in the standard form.

Complex Conjugates

Complex conjugates are pairs of complex numbers that have the same real part but opposite imaginary parts.

For instance, if you have a complex number \(a + bi\), its complex conjugate would be \(a - bi\). In practice, multiplying a complex number by its conjugate is useful because it results in a real number. This happens because:

• The imaginary parts cancel each other out during multiplication.
• This results in an expression without imaginary components.

In fact, when we multiplied \((-7-i)\) by its conjugate \((-7+i)\), the expression \(-7i + 7i\) vanishes, simplifying further to a real value. Thus, such operations are often used to move from a complex expression to a standard form, primarily when dividing complex numbers or simplifying expressions.

Standard Form of Complex Numbers

The standard form of a complex number is expressed as \(a + bi\) where \(a\) is the real part, and \(bi\) is the imaginary part.

When you perform operations such as addition, subtraction, multiplication, or division of complex numbers, it is common practice to convert the results back into this form. The standard form makes it easier to identify and utilize the real and imaginary components in further calculations or applications.

In the exercise above, the result of multiplying two complex conjugates resulted in a real number. After simplifying through cancellation and evaluation, we derived the real number 48, which technically can be expressed as \(48 + 0i\). However, when the imaginary part is zero, it is customary to simply write the real number. Therefore, standard form helps provide a clear and concise way of representing the outcome of complex arithmetic operations.