Problem 100 - Algebra and Trigonometry Real Mathematics, Real People

Question

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

Step-by-Step Solution

Answer

The expanded form of the complex number \(\left(-\frac{1}{3}+\frac{\sqrt{3}}{3} i\right)^{3}\) is \(-\frac{10}{27} + \frac{2\sqrt{3}}{27} i\).

Step 1: Express the binomial

Write the expression \(\left(-\frac{1}{3}+\frac{\sqrt{3}}{3} i\right)^{3}\) as \(\left(-\frac{1}{3}\right)^{3} + 3\left(-\frac{1}{3}\right)^{2}\left(\frac{\sqrt{3}}{3} i\right) + 3\left(-\frac{1}{3}\right)\left(\frac{\sqrt{3}}{3} i\right)^{2} + \left(\frac{\sqrt{3}}{3} i\right)^{3}\) using the Binomial Theorem.

Step 2: Simplifying fractions

Simplify the fractions in each term. \(-\frac{1}{27} + \frac{\sqrt{3}}{9} i - \frac{1}{9} - \frac{\sqrt{3}}{27} i = -\frac{10}{27} + \frac{2\sqrt{3}}{27} i\)

Step 3: Rewriting the imaginary term

Rewrite the terms having \(i^{2}\), using the elementary relation \(i^{2} = -1\) and further simplify the expression.

Step 4: Final Simplification

Finally, after substitution of \(i^{2} = -1\), simplify the entire expression to obtain the expanded complex number.

Key Concepts

Complex Numbers

A complex number consists of two parts: a real part and an imaginary part. The standard form is often written as \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part. This notation helps us understand complex numbers as an extension of the real numbers. They can be thought of as points or vectors in a two-dimensional space, with the x-axis representing the real part and the y-axis the imaginary part.

• A real number is a complex number where the imaginary part is zero (e.g., \(5 + 0i\)).
• An imaginary number has zero as the real part and any non-zero number as the imaginary part (e.g., \(0 + 3i\)).

Understanding complex numbers allows us to solve equations that don't have solutions solely in the realm of real numbers, such as \(x^2 + 1 = 0\). Using complex numbers, this equation has solutions \(x = i\) and \(x = -i\), where \(i\) is the imaginary unit.

Polynomial Expansion

Polynomial expansion is a mathematical process used to express a power of a binomial in its expanded form. For example, when expanding \((a + b)^n\), we use the Binomial Theorem:
\[(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\]
Here, \(\binom{n}{k}\) is a binomial coefficient, representing the number of ways to choose \(k\) successes in \(n\) trials, computed as \(\frac{n!}{k!(n-k)!}\). Each term in the expansion is found by multiplying these coefficients with powers of \(a\) and \(b\).

• It allows simplification of expressions raised to high powers, breaking them down into smaller, more manageable pieces.
• Using the Binomial Theorem, we can deal with both real and complex numbers, providing an indispensable tool for higher-degree polynomial calculations.

Imaginary Unit

The imaginary unit is designated as \(i\), defined by the property that \(i^2 = -1\). It forms the foundation for working with complex numbers and enables mathematicians and scientists to solve and express solutions to equations that involve square roots of negative numbers.

• \(i\) is crucial in extending our number system beyond real numbers, expanding our ability to solve all polynomial equations.
• When calculating expressions involving \(i\), remember that higher powers of \(i\) follow a repetitive pattern:
  - \(i^1 = i\)
  - \(i^2 = -1\)
  - \(i^3 = -i\)
  - \(i^4 = 1\)
  
• This cycle repeats every four powers, which helps simplify expressions where powers of \(i\) exceed this range.

The imaginary unit changes the landscape of algebra by allowing square roots of negative numbers, thus providing real solutions to otherwise unsolvable problems.