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Chapter 8: Problem 69

$$ \text { For Exercises 65-76, simplify and express in standard form. } $$ $$ (5-2 i)^{2} $$

Short Answer

Expert verified
The simplified expression in standard form is \(21 - 20i\).

Step by step solution

01

Set up the expression

We are asked to simplify the expression \((5-2i)^2\). To start, we write it as the square of a binomial: \((5 - 2i)(5 - 2i)\).
02

Apply the distributive property

Now, apply the distributive property to expand \((5 - 2i)(5 - 2i)\). First, multiply: \((5)(5) = 25\) \((5)(-2i) = -10i\) \((-2i)(5) = -10i\) \((-2i)(-2i) = 4i^2\) (Remember that \(i^2 = -1\); hence \(4i^2\) becomes \(4(-1) = -4\)).
03

Combine like terms

Now combine all the terms from the expansion: \[25 - 10i - 10i - 4\] First, add the real components: \(25 - 4 = 21\). Next, combine the imaginary components: \(-10i - 10i = -20i\).
04

Write the result in standard form

The standard form for a complex number is \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. So in this expression, the standard form is: \[21 - 20i\].

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Key Concepts

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

Binomial Expansion
When dealing with expressions like \((a - b)^2\), we talk about binomial expansion. A binomial is simply an expression involving two terms, and the expansion refers to multiplying it out fully. In our exercise, we start with the binomial \((5 - 2i)^2\). Instead of calculating directly, we rewrite the expression as \((5 - 2i)(5 - 2i)\), which makes it easier to expand using multiplication.
  • The first step is to apply the expression twice, based on the binomial theorem, such as \((a - b)(a - b)\), and calculate it as \(a^2 - 2ab + b^2\).
  • This helps in breaking down the expression into smaller, more manageable calculations.
By expanding the binomial, you simplify the process of finding the result. This approach is essential in algebra for working efficiently with polynomials and complex expressions.
Distributive Property
A vital tool in multiplication within algebra, the distributive property helps to simplify expressions. It's defined as \(a(b + c) = ab + ac\) and applies to multiplication across terms within parentheses. In our example, we use it to expand \((5 - 2i)^2\).
  • The distributive property enables multiplying each term inside the parentheses separately.
  • This means calculating \(5 \cdot 5, 5 \cdot (-2i), (-2i) \cdot 5,\) and \((-2i) \cdot (-2i)\).
  • It systematically breaks down complex multiplications into simpler parts, leading us to a complete expansion.
By using this property, we ensure that no terms are missed, and it offers a structured way to handle algebraic expressions. Ultimately, it simplifies a mathematical problem by transforming it into smaller, additive steps.
Imaginary Unit
The imaginary unit \(i\) forms the basis of complex numbers. Defined by the property \(i^2 = -1\), this allows us to handle square roots of negative numbers. In our scenario, when expanding \((5 - 2i)^2\), we encounter the term \((-2i) \cdot (-2i)\).
  • Normally, \((-2)(-2)= 4\), but because of the presence of \(i\), this becomes \(4i^2\).
  • \(i^2\)'s property transforms \(4i^2\) into \(4(-1)\), simplifying to \(-4\).
  • This conversion is crucial to obtaining the correct real number during multiplication.
Understanding \(i\) aids in managing complex-number arithmetic, as it turns irrational terms into manageable components, providing a full picture of how both real and imaginary parts interact in a complex number. This imaginary unit makes it possible for real numbers to coexist with their imaginary counterparts, creating complex numbers like \(21 - 20i\), where real and imaginary parts are visibly separate but function together.

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Most popular questions from this chapter

Bicycle Racing. A boy on a bicycle racing around an oval track has a position given by the equations \(x=-100 \sin \left(\frac{t}{4}\right)\) and \(y=75 \cos \left(\frac{t}{4}\right)\), where \(x\) and \(y\) are the horizontal and vertical positions in feet relative to the center of the track \(t\) seconds after the start of the race. Find out how long it takes the boy to complete one lap.

For Exercises 47 and 48 , refer to the following: Modern amusement park rides are often designed to push the envelope in terms of speed, angle, and ultimately \(g\)-force, and usually take the form of gargantuan roller coasters or skyscraping towers. However, even just a couple of decades ago, such creations were depicted only in fantasy-type drawings, with their creators never truly believing their construction would become a reality. Nevertheless, thrill rides still capable of nauseating any would-be rider were still able to be constructed; one example is the Calypso. This ride is a not-too-distant cousin of the better-known Scrambler. It consists of four rotating arms (instead of three like the Scrambler), and on each of these arms, four cars (equally spaced around the circumference of a circular frame) are attached. Once in motion, the main piston to which the four arms are connected rotates clockwise, while each of the four arms themselves rotates counterclockwise. The combined motion appears as a blur to any onlooker from the crowd, but the motion of a single rider is much less chaotic. In fact, a single rider's path can be modeled by the following graph: The equation of this graph is defined parametrically by $$ \begin{aligned} &x(t)=A \cos t+B \cos (-3 t) \\ &y(t)=A \sin t+B \sin (-3 t), 0 \leq t \leq 2 \pi \end{aligned} $$ Amusement Rides. What is the location of the rider at \(t=0, t=\frac{\pi}{2}, t=\pi, t=\frac{3 \pi}{2}\), and \(t=2 \pi\) ?

In Exercises 1-20, graph the curve defined by the following sets of parametric equations. Be sure to indicate the direction of movement along the curve. $$ x=t^{2}, y=t^{3}, t \text { in }[-2,2] $$

For Exercises 47 and 48 , refer to the following: Modern amusement park rides are often designed to push the envelope in terms of speed, angle, and ultimately \(g\)-force, and usually take the form of gargantuan roller coasters or skyscraping towers. However, even just a couple of decades ago, such creations were depicted only in fantasy-type drawings, with their creators never truly believing their construction would become a reality. Nevertheless, thrill rides still capable of nauseating any would-be rider were still able to be constructed; one example is the Calypso. This ride is a not-too-distant cousin of the better-known Scrambler. It consists of four rotating arms (instead of three like the Scrambler), and on each of these arms, four cars (equally spaced around the circumference of a circular frame) are attached. Once in motion, the main piston to which the four arms are connected rotates clockwise, while each of the four arms themselves rotates counterclockwise. The combined motion appears as a blur to any onlooker from the crowd, but the motion of a single rider is much less chaotic. In fact, a single rider's path can be modeled by the following graph: The equation of this graph is defined parametrically by $$ \begin{aligned} &x(t)=A \cos t+B \cos (-3 t) \\ &y(t)=A \sin t+B \sin (-3 t), 0 \leq t \leq 2 \pi \end{aligned} $$ Amusement Rides. Suppose the ride conductor was rather sinister and speeded up the ride to twice the speed. How would you modify the parametric equations to model such a change? Now vary the values of \(A\) and \(B\). What do you conjecture these parameters are modeling in this problem?

In Exercises 21-36, each set of parametric equations defines a plane curve. Find an equation in rectangular form that also corresponds to the plane curve. $$ x=t, y=\sqrt{t^{2}+1} $$
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