Problem 27 - Precalculus: Functions and Graphs

Question

Write the expression in the form \(a+b i,\) where \(a\) and \(b\) are real numbers. $$(2+5 i)^{3}$$

Step-by-Step Solution

Answer

The expression is \(-142 - 65i\).

Step 1: Expand the cube using the binomial theorem

Use the binomial theorem to expand \((2 + 5i)^3 = (2 + 5i)(2 + 5i)(2 + 5i) = \)\[ \sum_{k=0}^{3} \binom{3}{k} (2)^{3-k} (5i)^k. \]

Step 2: Evaluate each term of the expansion

Calculate each term individually:- When \(k = 0\), the term is \(1 \times 2^3 \times (5i)^0 = 8\)- When \(k = 1\), the term is \(3 \times 2^2 \times (5i)^1 = 60i\)- When \(k = 2\), the term is \(3 \times 2^1 \times (5i)^2 = -150\) (since \(i^2 = -1\))- When \(k = 3\), the term is \(1 \times 2^0 \times (5i)^3 = -125i\) (since \(i^3 = -i\))

Step 3: Combine all the terms

Add up all the components from each term:1. Real components: \(8 - 150 = -142\)2. Imaginary components: \(60i - 125i = -65i\)

Step 4: Write the final expression in the form \(a+bi\)

Combine the results from Step 3 to get the final answer: \(-142 - 65i\).

Key Concepts

Binomial Theorem

The binomial theorem is a powerful tool in mathematics used for expanding expressions that are raised to any power. It's particularly useful for expressions like \((a + b)^n\), where you need to expand the expression and find out the terms for each power of \(n\). This theorem simplifies the expansion by telling us exactly how to calculate each term in the expansion. The formula for the binomial theorem is:
\[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k}a^{n-k}b^k. \]
Here's what these parts mean:

• \(\sum\) stands for "sum" and means that you sum up all the terms from \(k = 0\) to \(k = n\).
• \(\binom{n}{k}\) is a binomial coefficient, also known as "n choose k", which tells us how many ways we can choose \(k\) elements from \(n\) elements.
• \(a^{n-k}\) and \(b^k\) are your terms \(a\) and \(b\) raised to the specified powers for that term.

Using the binomial theorem helps reduce the complexity of calculations in manual expansions, especially when dealing with terms involving imaginary numbers which follow different arithmetic rules than real numbers.

Imaginary Unit

The imaginary unit, denoted as \(i\), is a fundamental concept in complex numbers. It is defined by the property that \(i^2 = -1\). This definition allows us to extend the real number system to complex numbers.
When dealing with powers of \(i\), it's helpful to remember these basic identities:

• \(i^1 = i\)
• \(i^2 = -1\)
• \(i^3 = -i\)
• \(i^4 = 1\)

These repeat in a cycle every four powers, so it’s useful for simplifying expressions that include higher powers of \(i\).
The imaginary unit is not just a theoretical concept but is used extensively in fields like electrical engineering, quantum physics, and applied mathematics to represent phenomena that are not easily described using only real numbers.

Complex Number Operations

Complex numbers, of the form \(a + bi\), where \(a\) and \(b\) are real numbers, require unique operations because they include an imaginary part. Regarding operations, addition, subtraction, multiplication, and division follow specific rules.
Addition/Subtraction:

• Combine like terms by adding/subtracting the real parts and the imaginary parts separately.

Multiplication:

• Use the distributive property (like the FOIL method in algebra) to expand, then apply \(i^2 = -1\) wherever necessary.

Division:

• Multiply the numerator and the denominator by the conjugate of the denominator to eliminate the imaginary part of the denominator, making it real.

Mastering these operations allows you to manipulate complex numbers just like you do with real numbers. In the context of the provided exercise, combining like terms using these operations let us convert the expanded complex expression into a neat form \(a + bi\).