Problem 78 - Precalculus with Limits

Question

In Exercises 73 - 78, use the Binomial Theorem to expand the complex number. Simplify your result. \( \left(5 - \sqrt{3}i\right)^4 \)

Step-by-Step Solution

Answer

\((5 - \sqrt{3}i)^4 = 184 - 480\sqrt{3}i\.

Step 1: Applying the Binomial Theorem

First, use the Binomial Theorem to expand \((5 - \sqrt{3}i)^4\). This will result in a sum of terms. \begin{align*} (5 - \sqrt{3}i)^4 &= {4\choose 0}5^{4-0} (-\sqrt{3}i)^0 + {4\choose 1}5^{4-1} (-\sqrt{3}i)^1 + {4\choose 2}5^{4-2} (-\sqrt{3}i)^2 + {4\choose 3}5^{4-3} (-\sqrt{3}i)^3 + {4\choose 4}5^{4-4} (-\sqrt{3}i)^4 \\ &= 625 - 4 * 125 \sqrt{3}i - 6 * 25 * 3 + 4 * 5 \sqrt{3}i + 9 \end{align*}

Step 2: Simplification of the Result

Next, simplify the sum of terms obtained in previous step. \begin{align*} 625 - 4 * 125 \sqrt{3}i - 6 * 25 * 3 + 4 * 5 \sqrt{3}i + 9 &= 625 - 500\sqrt{3}i - 450 + 20\sqrt{3}i + 9 \\ &= 184 - 480\sqrt{3}i \end{align*} Thus, \((5 - \sqrt{3}i)^4 = 184 - 480\sqrt{3}i\).

Key Concepts

Complex Numbers

Complex numbers are an extension of the real numbers and are of the form \( a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit with the property that \( i^2 = -1 \). They allow for the solution of equations that have no solutions within the set of real numbers, such as \( x^2 + 1 = 0 \). When working with complex numbers, it is important to remember that the imaginary unit \( i \) behaves according to its defined property, and this affects operations like addition, subtraction, multiplication, and especially exponentiation.

In the exercise provided, \( (5 - \(\)i)^4 \), the number \( 5 - \(\)i \) is a complex number. Its expansion requires careful handling of the imaginary unit during the application of the Binomial Theorem. As the power increases, the imaginary unit's exponent should be simplified using the fact that \( i^2 = -1 \), \( i^3 = -i \), and \( i^4 = 1 \), and these patterns repeat cyclically.

Polynomial Expansion

Polynomial expansion is the process of expressing a polynomial that has been raised to a power as a sum of terms. The Binomial Theorem is a quick way to expand binomials raised to a power without having to multiply the binomial by itself repeatedly. According to the theorem, \( (a + b)^n \) can be expanded into a sum of terms of the form \( {n\choose k}a^{n-k}b^k \), where \( k \) ranges from 0 to \( n \), and \( {n\choose k} \) represents the binomial coefficient.

In the provided exercise, we expand \( (5 - \(\)i)^4 \) using the Binomial Theorem. Each term in the expansion corresponds to a specific choice of \( k \), and a mixture of powers of 5 and \( \(\)i \), multiplied by the appropriate binomial coefficient. The expansion process generates a polynomial where each term has to be simplified considering the properties of complex numbers to reach the final simplified form.

Simplifying Expressions

Simplifying expressions is the act of reducing a mathematical expression to its simplest form. This often means combining like terms, reducing fractions, or applying algebraic rules to make the expression easier to understand or work with. When dealing with the outcome of a polynomial expansion involving complex numbers, simplification includes handling the imaginary unit powers and combining real and imaginary parts separately.

In the solution's final step, the terms in \( 625 - 500\sqrt{3}i - 450 + 20\sqrt{3}i + 9 \) are combined by grouping the real and imaginary components. The real parts \( 625 \), \( -450 \), and \( 9 \) add up, while the imaginary terms \( -500\sqrt{3}i \) and \( 20\sqrt{3}i \) are combined, resulting in a simplified complex number expression. It is essential to proceed carefully, ensuring the coefficients of \( i \) are handled correctly and that all like terms are appropriately grouped, yielding a final expression that accurately represents the expanded and simplified complex polynomial.