Problem 97 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the Binomial Theorem to expand the complex number. Simplify your result. (Remember that \(i=\sqrt{-1 .})\) \((4+\sqrt{3} i)^{4}\)

Step-by-Step Solution

Answer

The simplified form of the complex number expansion \((4+\sqrt{3} i)^{4}\) is \(-704 - 576\sqrt{3}i\)

Step 1: Establish the Binomial Theorem formula

The Binomial Theorem formula is given as \((a+b)^n = \sum_{k=0}^n {n \choose k} a^{n-k}*b^k\). In this case, \(n = 4\), \(a = 4\), and \(b = \sqrt{3}i\).

Step 2: Substitution into the binomial expansion

Now we substitute the values of \(a\), \(b\), and \(n\) into the formula. This gives \((4+\sqrt{3} i)^{4} = \sum_{k=0}^{4} {4 \choose k} 4^{4-k}*(\sqrt{3}i)^k\).

Step 3: Calculate the binomial coefficients and powers

We then individually calculate the binomial coefficients and powers for each term: \( {4 \choose 0} 4^{4}*(\sqrt{3}i)^0\), \( {4 \choose 1} 4^{3}*(\sqrt{3}i)^1\), \( {4 \choose 2} 4^{2}*(\sqrt{3}i)^2\), \( {4 \choose 3} 4^{1}*(\sqrt{3}i)^3\) and \( {4 \choose 4} 4^{0}*(\sqrt{3}i)^4\).

Step 4: Calculate the sum of the terms

Add all the terms together. In doing this, don't forget that \(i^2 = -1\), \(i^3 =-i \), and \(i^4 = 1 \).

Step 5: Simplify the sum

Simplify this sum to get the final result.

Key Concepts

Complex Numbers

Understanding complex numbers is essential when diving into the world of more advanced algebraic concepts, such as the Binomial Theorem in this context. A complex number consists of two parts:

• The real part, for example, 4.
• The imaginary part, which involves the imaginary unit \(i\), such as \(\sqrt{3}i\).

The imaginary unit \(i\) is defined as \(i = \sqrt{-1}\). This definition is crucial, as it allows us to handle expressions involving the square roots of negative numbers, which are not otherwise possible with real numbers alone. When dealing with complex numbers in expansions, the imaginary unit \(i\) plays a pivotal role, especially when calculating powers, like \(i^2 = -1\) and beyond.

Expansion

In algebra, expansion is a method used to simplify expressions raised to a power. With the Binomial Theorem, we expand power expressions into a sum of terms. This makes complex expressions like \((4+\sqrt{3} i)^{4}\) more manageable. To apply the expansion:

• Identify the values of \(a\), \(b\), and \(n\) from the binomial expression \((a+b)^n\).
• Substitute these values into the Binomial Theorem formula: \((a+b)^n = \sum_{k=0}^n {n \choose k} a^{n-k}*b^k\).

This turns a complex structure into a sum of clearer smaller expressions which can be tackled individually. Once each term is evaluated, the result simplifies the originally complex binomial expression.

Binomial Coefficients

The Binomial Theorem involves coefficients called binomial coefficients, represented as \({n \choose k}\). These coefficients are derived from the numbers in Pascal's Triangle and help determine the relative weights of each term in an expansion. When expanding \((4 + \sqrt{3} i)^4\), binomial coefficients help calculate the coefficients of terms after expansion:

• \({4 \choose 0} = 1\)
• \({4 \choose 1} = 4\)
• \({4 \choose 2} = 6\)
• \({4 \choose 3} = 4\)
• \({4 \choose 4} = 1\)

These coefficients multiply with the individual term calculations, ensuring each term’s contribution is accurately reflected in the final expanded expression. Mastery of calculating these coefficients is essential for successful use of the Binomial Theorem.

Powers of i

The powers of \(i\), the imaginary unit, are an intriguing aspect of working with complex numbers. Each power of \(i\) follows a repeating cycle:

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

For powers \(i\) greater than 4, the cycle repeats. Knowing these powers allows you to simplify terms efficiently. For example, when expanding \((4 + \sqrt{3} i)^4\), calculating \((\sqrt{3}i)^k\) involves determining the power of \(i\) in each term. Proper application of these principles ensures a smooth process in reducing terms and reaching the final simplified form of a binomial expansion involving complex numbers.