Problem 101 Use the Binomial Theorem to expa... [FREE SOLUTION]

Chapter 9: Problem 101

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

Short Answer

Expert verified

The expansion of the given complex number using the Binomial Theorem is \(-\frac{1}{8} - \frac{\sqrt{3}+\sqrt{27}}{64}i\).

Step by step solution

Application of the binomial theorem

The binomial theorem states that \( (a + b)^n = \sum_{k=0}^{n} C(n, k) a^{n-k} b^k \), where \(C(n, k)\) are the binomial coefficients or combinations. We apply this theorem to \(\left(\frac{1}{4}-\frac{\sqrt{3}}{4} i\right)^{3}\).

Substituting values into the binomial theorem

In our case, \(a = \frac{1}{4}\), \(b = -\frac{\sqrt{3}}{4} i\), and \(n = 3\). If we substitute these values in the Binomial Theorem, the expression becomes \( = C(3,0) * (\frac{1}{4})^{3} * (-\frac{\sqrt{3}}{4}i)^{0} + C(3,1) * (\frac{1}{4})^{2} * (-\frac{\sqrt{3}}{4}i)^{1} + C(3,2) * (\frac{1}{4})^{1} * (-\frac{\sqrt{3}}{4}i)^{2} + C(3,3) * (\frac{1}{4})^{0} * (-\frac{\sqrt{3}}{4}i)^{3}\).\(C(3, 0)\), \(C(3, 1)\), \(C(3, 2)\), and \(C(3, 3)\) are combination values, and their values are 1, 3, 3, and 1 respectively.

Calculate using combination values

If we substitute these combination values and calculate, the above expression becomes \( = 1 * (\frac{1}{64}) * 1 + 3 * (\frac{1}{16}) * (-\frac{\sqrt{3}}{16}i) + 3 * (\frac{1}{4}) * (\frac{3}{16}) + 1 * 1 * (-\frac{\sqrt{27}}{64}i) = \frac{1}{64} - \frac{\sqrt{3}}{16}i - \frac{9}{64} -\frac{\sqrt{27}}{64}i\).

Simplify the expression

Combining the real parts and the imaginary parts gives the final simplified expression as \(-\frac{8}{64} - \frac{\sqrt{3}+\sqrt{27}}{64}i=-\frac{1}{8} - \frac{\sqrt{3}+\sqrt{27}}{64}i\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Complex Numbers

Complex numbers are numbers that have two parts: a real part and an imaginary part. They are typically written in the form of \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. The imaginary part involves the imaginary unit \(i\), where \(i^2 = -1\). This allows for mathematical solutions to equations that do not have real number solutions, like the square root of a negative number.

The real part is straightforward; it is similar to the numbers we typically use in arithmetic.
The imaginary part allows us to handle complex mathematical situations.

Understanding complex numbers is crucial in fields such as engineering and physics, where they are used to represent phenomena like electrical currents and waves.

Binomial Coefficients

Binomial coefficients are the numerical factors that multiply the terms in the expansion of a binomial raised to a power. In the binomial theorem, these are represented as \(C(n, k)\) or \(\binom{n}{k}\), and they represent combinations or ways to select items.

They are calculated using the formula: \(C(n, k) = \frac{n!}{k!(n-k)!}\), where \(!\) denotes a factorial.
In a binomial expansion, the binomial coefficients give you the weight of each term in the expansion.

For example, in the original exercise, the coefficients for \((a-b)^3\) are found using \(C(3, 0), C(3, 1), C(3, 2), C(3, 3)\), giving the sequence \(1, 3, 3, 1\), crucial for expanding the expression correctly.

Expansion

The expansion of a binomial expression is the process of expressing it as a sum of terms using powers of its components. The Binomial Theorem is the tool that facilitates the expansion. For \((a + b)^n\), the theorem allows us to expand this expression into a series:\[(a + b)^n = \sum_{k=0}^{n} C(n, k) a^{n-k} b^k\]where each term involves binomial coefficients.

Each successive term involves a decrease in the power of \(a\) and an increase in the power of \(b\).
In the given exercise, the complex number expression is expanded using the above-mentioned binomial pattern.

It's crucial to keep track of each term and its coefficient to ensure the expansion is performed accurately.

Imaginary Unit i

The imaginary unit \(i\) is the cornerstone of complex numbers, defined as \(i = \sqrt{-1}\). It introduces a system that extends beyond the real number line.

The rules for powers of \(i\) are cyclic: \(i^1 = i\), \(i^2 = -1\), \(i^3 = -i\), \(i^4 = 1\), and they repeat every four powers.
When expanding expressions that include \(i\), it's important to recognize these cyclical powers.

In the original exercise, we dealt with an expression involving \(i\). Keeping track of \(i\)'s power in each term helps avoid errors in the calculation and leads to the simplification of the problem.

91影视

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

Short Answer

Step by step solution

Application of the binomial theorem

Substituting values into the binomial theorem

Calculate using combination values

Simplify the expression

Key Concepts

Complex Numbers

Binomial Coefficients

Expansion

Imaginary Unit i

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