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Chapter 8: Problem 73

$$ \text { For Exercises 65-76, simplify and express in standard form. } $$ $$ (3+i)^{3} $$

Short Answer

Expert verified
The simplified expression is \(18 + 26i\).

Step by step solution

01

Identify the expression

The expression given is \((3+i)^3\), which needs to be simplified and expressed in the standard form \(a + bi\).
02

Use the binomial theorem

The binomial theorem states that \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\). We will apply this to expand \((3+i)^3\).
03

Expand the expression

Using the binomial expansion: \[(3+i)^3 = \binom{3}{0}3^3i^0 + \binom{3}{1}3^2i^1 + \binom{3}{2}3^1i^2 + \binom{3}{3}3^0i^3\].
04

Calculate each term individually

1) \(\binom{3}{0}3^3i^0 = 27\)2) \(\binom{3}{1}3^2i^1 = 3 \times 9 \times i = 27i\)3) \(\binom{3}{2}3^1i^2 = 3 \times 3 \times (-1) = -9\)4) \(\binom{3}{3}3^0i^3 = i^3 = -i\)
05

Combine like terms

Combine the real and imaginary parts:Real: \(27 - 9 = 18\)Imaginary: \(27i - i = 26i\)Thus, the expression simplifies to \(18 + 26i\).
06

Write the simplified expression in standard form

The expression \((3+i)^3\) simplified is \(18 + 26i\), which is in the standard form \(a + bi\).

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Key Concepts

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

Binomial Theorem
The Binomial Theorem is a powerful tool for expanding expressions raised to a power, particularly when dealing with expressions like
  • .
  • where:
The binomial coefficients \[ \binom{n}{k} = \frac{n!}{k!(n-k)!}\]can be calculated for each term in the expansion.
When applied to \(3 + i\),the expression becomes:
  • \( (3+i)^3 = \sum_{k=0}^{3} \binom{3}{k} 3^{3-k} i^k \)
This expansion method allows us to break down complex expressions into manageable parts.
Simplifying Expressions
Simplifying expressions involves reducing them to their simplest form by combining like terms. When simplifying an expression like \((3+i)^3\), we follow these steps:
  • Calculate each term from the binomial expansion.
  • Use the identity \(i^2 = -1\) to simplify powers of \(i\).
  • Combine real numbers with real numbers and imaginary numbers with imaginary ones.
For example, using the steps:
  • \(3^3 = 27\)
  • \(3^2 \times i = 27i\)
  • \(3 \times i^2 = -9\)
  • \(i^3 = -i\)
Finally, combine results:
  • Real: \(27 - 9 = 18\)
  • Imaginary: \(27i - i = 26i\)
Thus, after combining:\( (3+i)^3 = 18 + 26i \).This approach keeps mathematical expressions straightforward, ensuring simpler calculations.
Standard Form of Complex Numbers
Complex numbers are expressed in standard form as \(a + bi\),where
  • \(a\) is the real part
  • \(b\) is the imaginary part
For instance, in our previous calculation,
  • the standard form is \(18 + 26i\)
Representing complex numbers in this form enables easier addition, subtraction, and comparison with other complex numbers.
When multiplying or raising complex numbers to a power, always aim to revert back to this standard form. It simplifies interpretation and application in further calculations or problem-solving tasks. Using the standard form ensures clarity and precision in any complex arithmetic process.

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Most popular questions from this chapter

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. $$ x=2 \sin (3 t), y=3 \cos (2 t), t \text { in }[0,2 \pi] $$

Algebraically, find the polar coordinates \((r, \theta)\) where \(0 \leq \theta<2 \pi\) that the graphs \(r_{1}=1-2 \sin (2 \theta)\) and \(r_{2}=1-2 \cos (2 \theta)\) have in common.

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 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. $$ r^{2} \cos ^{2} \theta-r \sin \theta=-2 $$

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. $$ r=2+3 \sin \theta $$
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