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Chapter 8: Problem 76

$$ \text { For Exercises 65-76, simplify and express in standard form. } $$ $$ (4-3 i)^{3} $$

Short Answer

Expert verified
The simplified expression is \(172 - 117i\).

Step by step solution

01

Understanding the Expression

We are given the expression \((4-3i)^3\), where \(i\) is the imaginary unit such that \(i^2 = -1\). Our task is to simplify this expression and express it in standard form.
02

Expanding the Cubic Expression

To simplify \((4 - 3i)^3\), we can use the binomial theorem, which states \((a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3\). Here, \(a = 4\) and \(b = -3i\). Substituting these into the formula gives:\[(4 - 3i)^3 = 4^3 + 3 \times 4^2 \times (-3i) + 3 \times 4 \times (-3i)^2 + (-3i)^3.\]
03

Calculate Each Term

We compute each term from the expansion in Step 2:1. \(4^3 = 64\).2. \(3 \times 4^2 \times (-3i) = 3 \times 16 \times (-3i) = -144i\).3. \(3 \times 4 \times (-3i)^2 = 3 \times 4 \times (9)(-1) = 108\).4. \((-3i)^3 = -27i(-1) = 27i\).
04

Combine Like Terms

Combine the results from Step 3:\[64 - 144i + 108 + 27i.\]Simplify by combining like terms:- Combine the real numbers: \(64 + 108 = 172\).- Combine the imaginary numbers: \(-144i + 27i = -117i\).Thus, the expression simplifies to:\[172 - 117i\].
05

Verify and Express in Standard Form

The standard form for a complex number is \(a + bi\). We have the expression \(172 - 117i\), where \(a = 172\) and \(b = -117\). The expression is already in 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 crucial tool in algebra that allows us to expand powers of binomials. If you've got a binomial expression like
  • \((a + b)^n\), it can be expanded into a sum involving terms of the form \(\binom{n}{k} a^{n-k} b^k\).
This formula is particularly handy when you need to compute powers of sums without multiplying the expression repeatedly. The theorem is based on the binomial coefficients \(\binom{n}{k}\), which can be found in Pascal's Triangle or calculated using the formula:\[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \]These coefficients tell us how many ways we can choose \(k\) elements from a set of \(n\) elements. In our exercise, we use the theorem for
  • \((4-3i)^3\)
to simplify the expression easily.
Imaginary Unit
At the heart of complex numbers lies the imaginary unit \(i\). This little symbol is defined by the equation:
  • \(i^2 = -1\)
This strange characteristic is what enables us to handle square roots of negative numbers. Real numbers can't accomplish this, since there's no real number whose square is
  • negative.
The presence of \(i\) allows us to delve into complex numbers, providing solutions to equations that wouldn't be possible otherwise. These numbers appear in the form of
  • \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part.
Standard Form of a Complex Number
Expressing complex numbers in standard form is essential for clarity and precision. The standard form is described as
  • \(a + bi\)
where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit. The part \(a\) represents the real component, while \(bi\) represents the imaginary component.Consider an example, our previous exercise result,
  • \(172 - 117i\)
Here, \(172\) is the real part and \(-117i\) represents the imaginary part. This format is vital because it allows
  • easy comparison
  • addition
  • subtraction of complex numbers
Understanding the standard form can help you work effortlessly with complex numbers.

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Most popular questions from this chapter

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. $$ x=\sqrt{t-1}, y=\sqrt{t} $$

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