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Chapter 10: Problem 68

Multiply. Give answers in standard form. $$ (7+2 i)(7-2 i) $$

Short Answer

Expert verified
53

Step by step solution

01

- Apply the difference of squares formula

Recognize that (7+2i)(7-2i) is of the form (a+bi)(a-bi), which can be simplified using the formula (a^2 - (bi)^2).
02

- Square the first term

Calculate the square of the real part: 7^2 = 49.
03

- Square the imaginary term

Calculate the square of the imaginary part: (2i)^2 = 4i^2.
04

- Substitute and simplify

Substitute the squared terms into the formula, remembering that i^2 = -1: 49 - 4i^2 = 49 - 4(-1).
05

- Final simplification

Simplify the expression: 49 - 4(-1) = 49 + 4 = 53.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Difference of Squares
The difference of squares is a crucial concept in algebra. It states that for any two numbers, say \(a\) and \(b\), the expression \((a + b)(a - b)\) can be simplified to \(a^2 - b^2\). This formula is extremely helpful when multiplying complex numbers like in our exercise, \((7+2i)(7-2i)\).
To apply this concept, we see that:
  • \( a = 7 \)
  • \( b = 2i \)
Therefore, we write:
\[ \ (7+2i)(7-2i) = 7^2 - (2i)^2 \]
By simplifying each term separately, we make the process easier and more manageable.
Imaginary Unit
The imaginary unit, denoted as \(i\), is defined by its property that \(i^2 = -1\). This property is essential whenever you are dealing with complex numbers.
In our example, we see the imaginary part \(2i\). When squared, it behaves as follows:
  • \( (2i)^2 = 4i^2 \)
  • Since \(i^2 = -1\), it becomes:
\[ 4i^2 = 4(-1) = -4 \]
This negative sign is because the square of the imaginary unit flips the sign. Recognizing and applying this rule simplifies calculations significantly.
Standard Form
The standard form of a complex number is written as \(a + bi\), where \(a\) and \(b\) are real numbers. This format helps in clearly distinguishing between the real and imaginary parts.
In this exercise, our final step involves simplifying and ensuring the result is in standard form.
We start with: \[ 49 - 4(-1) \]
Simplify it:
\[ 49 + 4 = 53 \]
Since there is no imaginary component left, the final standard form is simply \(53\). This is now a purely real number.
Understanding how to put your answers in this form is key to successfully dealing with complex number arithmetic.

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