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Chapter 8: Problem 39

In Exercises 13-40, perform the indicated operation, simplify, and express in standard form. $$ (-i+17)(2+3 i) $$

Short Answer

Expert verified
The result is \(37 + 49i\).

Step by step solution

01

Distribute the First Term

The expression to simplify is \((-i + 17)(2 + 3i)\). Start by distributing the first term of the first binomial:\[-i \cdot (2 + 3i) = -i \cdot 2 - i \cdot 3i.\]This gives us:\[-2i - 3i^2.\]
02

Simplify Using i^2 Property

We know that \(i^2 = -1\), so substitute \(-3i^2\) with \(-3(-1)\):\[-2i + 3.\]
03

Distribute the Second Term

Next, distribute the second term of the first binomial:\[17 \cdot (2 + 3i) = 17 \cdot 2 + 17 \cdot 3i.\]This gives us:\[34 + 51i.\]
04

Combine Like Terms

Add the results from Steps 1 and 3 together:\[(-2i + 3) + (34 + 51i) = 3 + 34 + (-2i + 51i).\]Combine like terms:\[37 + 49i.\]
05

Express in Standard Form

The standard form of a complex number is \(a + bi\). From Step 4, we have:\[37 + 49i.\]This is already in standard form.

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Key Concepts

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

operations with complex numbers
When dealing with complex numbers, you will often need to perform operations such as addition, subtraction, multiplication, and division. Complex numbers are represented in the form \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. For multiplication, such as in the given exercise, it's important to apply the distributive property duly to ensure each term is multiplied correctly.
  • **Addition and Subtraction:** Simply combine like terms, add or subtract the real parts together, and do the same for the imaginary parts.

  • **Multiplication:** It's similar to multiplying binomials in algebra where you multiply each term of one complex number by each term of the other.

  • **Division:** Use the conjugate of the denominator to eliminate any imaginary unit in the denominator, simplifying your expression.

Understanding these operations is fundamental when working with any expression consisting of complex numbers.
distributive property
The distributive property is a key concept when multiplying expressions, especially in complex numbers. It states that you multiply each term in the first expression by every term in the second expression. In this exercise, the expression is \((-i + 17)(2 + 3i)\).To apply the distributive property effectively:
  • Multiply the first term of the first binomial by each of the terms in the second binomial. For instance, multiply \(-i\) by \(2\) and then by \(3i\).

  • Next, take the second term of the first binomial, here \(17\), and multiply by each term of the second binomial, \(2\) and \(3i\).

  • After performing these multiplications, you'll combine all your results.

  • Finally, collect and simplify like terms, often resulting in a simpler complex expression.

Efficiently using the distributive property simplifies the process of dealing with complex multiplication.
standard form of complex numbers
The standard form of a complex number is \(a + bi\), where \(a\) is the real component, and \(b\) is the imaginary component along with the imaginary unit \(i\). This standard form ensures clarity when expressing complex numbers.
  • **Real Part:** The real part \(a\) is the component without \(i\). In our expression after simplifying, this is 37.

  • **Imaginary Part:** The imaginary part \(b\) is the coefficient of \(i\). From our solved exercise, the coefficient is 49, expressed as \(49i\).

  • To express any complex number this way, ensure all terms are clearly attributed as either real or imaginary, avoiding confusion.

Expressing complex numbers in this form helps streamline presentation and analysis.
imaginary unit
The imaginary unit \(i\) is a fundamental component of complex numbers, invented to solve equations where negatives under square roots occur. By definition, \(i\) is the square root of \(-1\). It allows us to extend the real number system to include solutions to polynomial equations that have no real number solutions.Some key points about \(i\) are:
  • **\(i^2 = -1\):** This crucial property is used widely, as seen in the solution, to convert terms involving \(i^2\) into real numbers.

  • **Powers of \(i\):** These cycle in a predictable pattern: \(i^1 = i\), \(i^2 = -1\), \(i^3 = -i\), and \(i^4 = 1\). This repeats beyond \(i^4\), helping simplify expressions involving higher powers.

  • Complex numbers leverage \(i\) to blend real numbers with their imaginary counterparts, broadening the scope of calculations.

Incorporating \(i\) enables us to tackle problems previously deemed unsolvable within the realm of "real" solutions alone.

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Most popular questions from this chapter

Fan Blade. The position on the tip of a ceiling fan is given by the parametric equations \(x=\sin (10 t)\) and \(y=\cos (10 t)\), where \(x\) and \(y\) are the vertical and lateral positions relative to the center of the fan, respectively, and \(t\) is the time in seconds. How long does it take for the fan, blade to make one complete revolution?

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. $$ x=3 t, y=t^{2}-1 $$

For Exercises 73 and 74, refer to the following: The lemniscate motion occurs naturally in the flapping of birds' wings. The bird's vertical lift and wing sweep create the distinctive figure-eight pattern. The patterns vary with different wing profiles. Flapping Wings of Birds. Compare the two following possible lemniscate patterns by graphing them on the same polar graph: \(r^{2}=4 \cos (2 \theta)\) and \(r^{2}=\frac{1}{4} \cos (2 \theta)\).

In Exercises 45-68, graph each equation. In Exercises 63-68, convert the equation from polar to rectangular form first and identify the resulting equation as a line, parabola, or circle. $$ r^{2}=16 \sin (2 \theta) $$

For Exercises 75 and 76, refer to the following: Many microphone manufacturers advertise their exceptional pickup capabilities that isolate the sound source and minimize background noise. The name of these microphones comes from the pattern formed by the range of the pickup. Cardioid Pickup Pattern. Graph the cardioid curve to see what the range of a microphone might look like: \(r=-4-4 \sin \theta\).
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