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Chapter 9: Problem 73

Use the Binomial Theorem to expand the complex number. Simplify your result. $$(1+i)^{4}$$

Short Answer

Expert verified
The expanded and simplified form of the complex number \((1 + i)^4\) is 2.

Step by step solution

01

Write down the Binomial Theorem

The binomial theorem is used to expand the expression (a + b)^n into a sum of terms. It is summarised as follows: (a + b)^n = 危 (from k=0 to n) [n choose k] * a^(n-k) * b^k. Here, for our problem, a=1, b=i, and n=4.
02

Substitute the values into the Binomial Theorem

Substitute the given values into the Binomial Theorem. Plug a=1, b=i, and n=4 into the theorem. This gives us: (1 + i)^4 = 危 (from k=0 to 4) [4 choose k] * 1^(4-k) * i^k.
03

Simplify the equation

Simplify the equation. Remember that 1 raised to any power is 1, and i has a repeated cycle: i^1=i, i^2=-1, i^3=-i, i^4=1. So, simplification gives us: (1 + i)^4 = [4 choose 0]*i^0 + [4 choose 1]*i^1 + [4 choose 2]*i^2 + [4 choose 3]*i^3 + [4 choose 4]*i^4. This equals: 1*1 + 4*i - 6 - 4i + 1 = 2.
04

Write the final answer

After simplifying, we get the result that (1 + i)^4 = 2. This is the expanded form of the given complex number, simplified.

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Key Concepts

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

Complex Numbers
Complex numbers are numbers that have a real part and an imaginary part. They are usually written in the form \(a + bi\), where \(a\) is the real part, and \(b\) is the imaginary part. The imaginary part includes \(i\), which represents the square root of -1, i.e., \(i = \sqrt{-1}\). This makes complex numbers very different from standard real numbers.
  • Real part: a number without \(i\).
  • Imaginary part: a number with \(i\).
To work with complex numbers, you can use arithmetic operations just like with real numbers. These operations can include addition, subtraction, multiplication, and even exponentiation, like raising them to a power, as seen in our given exercise.
Polynomial Expansion
Polynomial expansion involves expressing an expression in terms of a sum of powers. The Binomial Theorem is especially helpful in expanding expressions that are raised to a power.When you see an expression like \((a + b)^n\), it expands into a sum of terms that can be calculated using the formula:\[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{(n-k)} b^k \]Let's break down the components:
  • \(\binom{n}{k}\): Represents the binomial coefficient or "n choose k," indicating the number of ways to choose \(k\) items from \(n\) items, mathematically expressed as \(\frac{n!}{k!(n-k)!}\).
  • \(a^{(n-k)}\): The part of the term contributed by \(a\).
  • \(b^k\): The part of the term contributed by \(b\).
In the exercise, you apply this theorem to expand \((1 + i)^4\) by substituting \(a = 1\), \(b = i\), and \(n = 4\). By following this method, each term in the expansion contributes to the final sum.
Powers of i
Understanding the powers of \(i\) is crucial when dealing with expressions involving complex numbers. The imaginary unit \(i\) has a unique property; when raised to powers, it follows a cyclic pattern:
  • \(i^1 = i\)
  • \(i^2 = -1\)
  • \(i^3 = -i\)
  • \(i^4 = 1\)
This cycle repeats every four powers. So for any positive integer \(n\), the value of \(i^n\) can be determined by dividing \(n\) by 4 and looking at the remainder:
  • If the remainder is 1, \(i^n = i\).
  • If the remainder is 2, \(i^n = -1\).
  • If the remainder is 3, \(i^n = -i\).
  • If the remainder is 0, \(i^n = 1\).
In the expansion \((1 + i)^4\), you evaluate the powers of \(i\) at each term in the binomial expansion, which is critical in simplifying the polynomial to the final result \(2\). Mastering the cycle of \(i\) allows quick simplifications in calculations involving complex numbers.

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Most popular questions from this chapter

Determine whether the statement is true or false. Justify your answer. The number of permutations of \(n\) elements can be determined by using the Fundamental Counting Principle.

A local college is forming a six-member research committee having one administrator, three faculty members, and two students There are seven administrators, 12 faculty members and 20 students in contention for the committee. How many six-member committees are possible?

Simplify the difference quotient, using the Binomial Theorem if necessary.\(\frac{f(x+h)-f(x)}{h}\). $$f(x)=\sqrt{x}$$

Determine whether the statement is true or false. Justify your answer. $$\sum_{i=1}^{4}\left(i^{2}+2 i\right)=\sum_{i=1}^{4} i^{2}+2 \sum_{i=1}^{4} i$$

In the Louisiana Lotto game, a player randomly chooses six distinct numbers from 1 to \(40 .\) In how many ways can a player select the six numbers?
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