Problem 17 - Precalculus

Question

In Exercises 14 - 17 , use Pascal's Triangle to simplify the given power of a complex number. $$ \left(\frac{\sqrt{2}}{2}-\frac{\sqrt{2}}{2} i\right)^{4} $$

Step-by-Step Solution

Answer

The simplified result is \(-1\).

Step 1: Write Pascal's Triangle Row

To simplify the complex number raised to a power, we use binomial expansion. For the power of 4, we use the fifth row (starting from 0) of Pascal's Triangle: 1, 4, 6, 4, 1. These numbers are the binomial coefficients.

Step 2: Set Up Binomial Expansion

Apply the binomial theorem: \( (a + b)^n = \sum_{k=0}^{n} C(n, k) a^{n-k} b^{k} \). Here, \( a = \frac{\sqrt{2}}{2} \), \( b = -\frac{\sqrt{2}}{2}i \), and \( n = 4 \). The expansion becomes: \((\frac{\sqrt{2}}{2})^{4} - 4(\frac{\sqrt{2}}{2})^{3}(\frac{\sqrt{2}}{2}i) + 6(\frac{\sqrt{2}}{2})^{2}(\frac{\sqrt{2}}{2}i)^{2} - 4(\frac{\sqrt{2}}{2})(\frac{\sqrt{2}}{2}i)^{3} + (\frac{\sqrt{2}}{2}i)^{4} \).

Step 3: Compute Individual Terms

Calculate each term separately: - \((\frac{\sqrt{2}}{2})^4 = \frac{1}{4}\)- \(-4(\frac{\sqrt{2}}{2})^3(\frac{\sqrt{2}}{2}i) = -4\frac{1}{4}i = -i\)- \(6(\frac{\sqrt{2}}{2})^2((\frac{\sqrt{2}}{2}i)^2 = 6\frac{1}{2}(-\frac{1}{2}) = -\frac{3}{2}\)- \(-4(\frac{\sqrt{2}}{2})(\frac{\sqrt{2}}{2}i)^3 = 4i\frac{1}{4}i = i\)- \((\frac{\sqrt{2}}{2}i)^4 = \frac{1}{4}\)

Step 4: Simplify Expression

Combine all the computed terms: \[\frac{1}{4} - i - \frac{3}{2} + i + \frac{1}{4} = \frac{1}{4} + \frac{1}{4} - \frac{3}{2} = -1\]. The complex terms cancel each other out (\(-i + i = 0\)).

Step 5: Final Result

The simplified result for \(\left(\frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} i\right)^{4}\) is \(-1\).

Key Concepts

Complex Numbers

Complex numbers are a combination of real and imaginary numbers. They are in the form \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part with \(i\) representing the square root of \(-1\). This number system was designed to solve equations that don't have solutions in the set of real numbers.
For example, the equation \(x^2 + 1 = 0\) doesn't have any real number solutions, but it has the complex solutions \(i\) and \(-i\). Complex numbers appear naturally in many areas of mathematics and engineering.
When handling complex numbers:

• You can add, subtract, and multiply them just like polynomials.
• Conjugation involves changing the sign of the imaginary part: \(a+bi\) becomes \(a-bi\).
• The modulus is given by \(|a+bi| = \sqrt{a^2 + b^2}\).
• They can be represented on a complex plane, with the x-axis being the real part and the y-axis being the imaginary part.

Binomial Expansion

Binomial expansion allows us to expand powers of binomials, expressions containing two terms. The expansion is written using binomial coefficients derived from Pascal's Triangle. For example, \((x + y)^4\) can be expanded using the fifth row of Pascal's Triangle: 1, 4, 6, 4, 1.
Each term in the expansion is composed of:

• A binomial coefficient from Pascal's Triangle.
• Factors of \(x\) and \(y\) raised to powers that sum up to the binomial power.

In practice:

• The term \((x + y)^4 = 1x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4\).
• Coefficients are calculated as \(C(n, k) = \frac{n!}{k! (n-k)!}\).

Understanding binomial expansion is vital for simplifying expressions, especially when dealing with powers of complex numbers or algebraic structures in calculus and beyond.

Binomial Theorem

The Binomial Theorem provides a formula for the expansion of binomial expressions raised to any power. Symbolically, it is expressed as \((a + b)^n = \sum_{k=0}^{n} C(n, k) a^{n-k} b^{k}\).
Key insights of the binomial theorem:

• It makes use of the binomial coefficients \(C(n, k)\).
• Each term combines powers of \(a\) and \(b\) such that the sum of the exponents equals \(n\).
• The theorem is particularly straightforward for positive integer exponents and each term in the expansion follows a predictable pattern.

For students and mathematicians, the binomial theorem is a powerful tool for tackling problems involving expansion which might not be easily solved by direct multiplication.

Power of a Complex Number

Raising a complex number to a power often requires strategic approaches like using the binomial theorem or polar representation. In many problems like the one given, breaking down the process into simple steps helps.
For instance, for \( \left(\frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2}i\right)^4\), utilize the binomial theorem and observe how terms simplify and often cancel.
Important points to remember:

• Use binomial expansion for powers of binomials. Complex terms in powers often lead to simplifications, especially when conjugate terms exist.
• When dealing with powers, calculate each term carefully, simplifying imaginary terms using \(i^2 = -1\).
• Remember that simplifying the power of complex numbers sometimes yields simple regular numbers, due to the nature of the imaginary components cancelling out.

Complex arithmetic becomes easier with these step-by-step approaches, as it combines logical methods with algebraic principles.