Problem 96 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the Binomial Theorem to expand the complex number. Simplify your result. (Remember that \(i=\sqrt{-1 .})\) \((5+\sqrt{-9})^{3}\)

Step-by-Step Solution

Answer

The simplified result of \((5+\sqrt{-9})^{3}\) is \(98+144i\).

Step 1: Interpret Complex Number

Firstly, observe that \(\sqrt{-9} = i \cdot \sqrt{9} = 3i\). Therefore, \(5+\sqrt{-9}\) can be rewritten as \(5+3i\).

Step 2: Apply Binomial Theorem

The Binomial Theorem states that \((a + b)^{n} = \sum_{k=0}^{n} \dbinom{n}{k} a^{n-k}b^{k}\). Apply the formula to \((5+3i)^3\).

Step 3: Expand Using Binomial Theorem

Expanding, we get \( \dbinom{3}{0}5^{3}(3i)^{0} + \dbinom{3}{1}5^{2}(3i)^{1} + \dbinom{3}{2}5^{1}(3i)^{2} + \dbinom{3}{3}5^{0}(3i)^{3}\). Calculate each term individually to get \(125 + 3 * 25 *3i - 3 * 5 * 9 - 27i\).

Step 4: Simplify the Result

Combine like terms resulting in \(125-27 + 225i -81i\) which simplifies to \(98+144i\).

Key Concepts

Complex Numbers

Complex numbers are numbers that have both a real part and an imaginary part. These numbers are incredibly useful in many areas of mathematics and engineering, especially when dealing with polynomial equations. A complex number is typically written in the form \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part. Understanding complex numbers involves recognizing that they can plot on a two-dimensional plane, with the real part on the x-axis and the imaginary part on the y-axis. This allows for a more versatile way of handling numbers and understanding phenomena that involve two dimensions.

Expansion

In mathematics, expansion refers to the process of expressing a mathematical expression in an extended form. In the context of complex numbers and the Binomial Theorem, expansion allows us to simplify expressions raised to a power, like \((5 + 3i)^3\). The Binomial Theorem is a powerful tool for expansion. This theorem provides a formula to expand expressions of the form \((a + b)^n\). Each term in the expansion is computed using the binomial coefficient \(\dbinom{n}{k}\), and involves raising each component \(a\) and \(b\) to the appropriate powers and multiplying by the corresponding coefficient.

Imaginary Unit

The imaginary unit, denoted as \(i\), is a fundamental concept in complex numbers. It is defined as \(i = \sqrt{-1}\). This concept allows us to extend the real number system to accommodate solutions to equations like \(x^2 = -1\), which has no real solution. Imaginary numbers are built on this unit, forming the basis for complex numbers altogether. When working with powers of \(i\), it's important to remember the cyclical pattern:

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

These powers repeat every four exponents, making calculations manageable when simplifying expressions involving \(i\).

Algebra Concepts

Algebra encompasses a variety of principles and techniques used to solve equations and understand relationships between values. Key algebra concepts include operations with numbers, variables, and the application of the Binomial Theorem, which is used for expanding expressions like \((5+3i)^3\). In this case, understanding how to manipulate complex numbers and apply the binomial expansion requires familiarity with arithmetic operations on complex numbers, including addition, subtraction, multiplication, and understanding how to simplify the results. Simplifying expressions is crucial in algebra, as it lets us rewrite equations in a form that is easier to solve or interpret, demonstrating why algebra is a fundamental skill in mathematics.