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Chapter 8: Problem 17

Simplify each expression to a single complex number. $$ (4-2 i)(4+2 i) $$

Short Answer

Expert verified
20

Step by step solution

01

Identify the expression type

The given expression \((4 - 2i)(4 + 2i)\) is in the form of \((a - b)(a + b)\), which is a difference of squares.
02

Apply the difference of squares formula

The difference of squares formula is \((a - b)(a + b) = a^2 - b^2\). Here, \(a = 4\) and \(b = 2i\).
03

Compute the square of 'a'

Calculate \(a^2 = 4^2 = 16\).
04

Compute the square of 'b'

Calculate \(b^2 = (2i)^2 = 4i^2\). Since \(i^2 = -1\), this becomes \(4(-1) = -4\).
05

Substitute and simplify

Substitute \(a^2 - b^2\) with the calculated values: \(16 - (-4) = 16 + 4 = 20\).

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Key Concepts

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

Difference of Squares
A powerful algebraic tool, the difference of squares formula, is commonly used to simplify expressions like
  • extit{(a - b)(a + b) = a^2 - b^2}
In the given expression,
  • example: extit{(4 - 2i)(4 + 2i)}
We notice it follows the pattern of
  • extit{a - b}
  • extit{a + b}
where
  • extit{a = 4}
  • extit{b = 2i}
By recognizing this structure, you can effortlessly substitute the squares into the formula, simplifying the expression.
This approach sheds complexity and delivers quicker results.
Imaginary Unit
Understanding the imaginary unit, denoted as
  • extit{i}
, is crucial when dealing with complex numbers. It is defined such that
  • extit{i}
equals the square root of
  • extit{-1}
This implies that when
  • extit{i}
is squared, it produces
  • extit{-1}
Understanding this property,
  • extit{i^2 = -1}
is essential because it allows you to transform complex expressions into simpler real numbers. For instance,
  • extit{(2i)^2 = 4i^2 = 4(-1) = -4}
This conversion facilitates straightforward substitution into specified formulas, such as the difference of squares.
Simplification of Expressions
Simplifying expressions isn't just about applying formulas. It often involves
  • extit{consistent recognition of patterns.
In this example,
  • extit{(4 - 2i)(4 + 2i)
Recognizing it as a difference of squares underlines the importance of boiling down complexity by leveraging known algebraic identities.
When you substitute
  • extit{a^2 = 16}
  • extit{b^2 = -4}
Into the formula
  • extit{a^2 - b^2},
It directs the computation down to basic arithmetic:
  • extit{16 - (-4) = 16 + 4 = 20}.
This procedure remarkably reduces otherwise arduous and lengthy calculations.

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Most popular questions from this chapter

A large boulder needs to be dragged to a new position. If pulled directly horizontally, the boulder would require 400 pounds of pulling force to move. We need to pull the boulder using a rope tied to the back of a large truck, forming a \(15^{\circ}\) angle from the ground. How much force will the truck need to pull with?

Sketch the parametric equations for \(-2 \leq t \leq 2\).$$ \left\\{\begin{array}{l} x(t)=1+2 t \\ y(t)=t^{2} \end{array}\right. $$

Compute each of the following, leaving the result in polar \(r e^{i \theta}\) form. $$ \left(2 e^{\frac{2 \pi}{3} i}\right)\left(4 e^{\frac{5 \pi}{3} i}\right) $$

Compute each of the following, leaving the result in polar \(r e^{i \theta}\) form. $$ \frac{24 e^{\frac{4 \pi}{3} i}}{6 e^{\frac{\pi}{2} i}} $$

Sketch a graph of the polar equation. $$ r=3+3 \sin (\theta) $$
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