91Ó°ÊÓ

Complex numbers are numbers that are expressed in the form \( a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit. The imaginary unit \( i \) has a special property: \( i^2 = -1 \). This allows complex numbers to be used in various mathematical calculations, providing a means to work with numbers that have an imaginary component.

When working with complex numbers in binomial expansion, like in this exercise, treat \( i \) just like a variable while applying its special powers. In the expression \((2-3i)^6\), both real and imaginary parts will interact, and powers of \( i \) will contribute to the final result. Always remember that different powers of \( i \) simplify differently:

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

These patterns repeat every four powers and help in simplifying expressions involving \( i \).

Binomial coefficients, denoted as \( C(n, k) \) or \( \binom{n}{k} \), play a central role in the Binomial Theorem. They determine the number of ways you can choose \( k \) elements from a set of \( n \) elements, and they are used to expand binomials raised to a power.

In the expression \((2-3i)^6\), the binomial coefficients are calculated for each term in the expansion. For expansion purposes, when expanding a binomial like \((a + b)^n\), each term is of the form:

\[ C(n, k) \cdot a^{n-k} \cdot b^k \]

The coefficients for \((2-3i)^6\) are:

• \(C(6,0) = 1\)
• \(C(6,1) = 6\)
• \(C(6,2) = 15\)
• \(C(6,3) = 20\)
• \(C(6,4) = 15\)
• \(C(6,5) = 6\)
• \(C(6,6) = 1\)

These coefficients multiply each corresponding term to construct the final expanded expression.

In mathematics, the powers of \( i \) are crucial when dealing with expressions involving complex numbers. The imaginary unit \( i \) simplifies different powers of itself in a repetitive cycle, and understanding this cycle is essential when expanding complex numbers.

As you expand a binomial expression involving \( i \), like \((2 - 3i)^6\), substituting appropriate powers of \( i \) into the expanded form is necessary:

• \(i^2 = -1\)
• \(i^3 = -i\)
• \(i^4 = 1\) (and starts repeating: \(i^5 = i, i^6 = -1\), and so on)

By remembering these simplifications, you can transform powers of \( i \) to their simplest forms and combine like terms. In this exercise, observing these patterns allows for the simplification of the expression into a manageable final result. This knowledge aids significantly in resolving complex expressions accurately.