Problem 42 - College Algebra with Corequisite Support

Question

For the following exercises, use a calculator to help answer the questions. Evaluate \((1+i)^{k}\) for \(k=4,8\) and 12 . Predict the value if \(k=16\).

Step-by-Step Solution

Answer

For \((1+i)^k\), the values are -4, 16, -64, and predict 256 for \(k=4,8,12,16\).

Step 1: Evaluate (1+i)^4

Calculate \[ (1+i)^4 = (1+i) \times (1+i) \times (1+i) \times (1+i) \]Using the binomial theorem, this equates to:\[ (1 + i)^4 = 1^4 + 4 \cdot 1^3 \cdot i + 6 \cdot 1^2 \cdot i^2 + 4 \cdot 1 \cdot i^3 + i^4 \]Calculating we get:\[ = 1 + 4i + 6(-1) + 4(-i) + 1 \]\[ = 1 + 4i - 6 - 4i + 1 \]\[ = -4 + 0i \] which simplifies to \(-4\).

Step 2: Evaluate (1+i)^8

Now calculate \[ (1+i)^8 = ((1+i)^4)^2 \]Using the result from Step 1,\[ = (-4)^2 = 16 \].

Step 3: Evaluate (1+i)^{12}

Calculate \[ (1+i)^{12} = ( (1+i)^4)^3 = (-4)^3 \]This simplifies to\[ = -64 \].

Step 4: Predict (1+i)^{16}

We observe the pattern: \( (1+i)^4 = -4 \), \( (1+i)^8 = 16 \), \( (1+i)^{12} = -64 \). Each step is multiplying by -4. Therefore:\[ (1+i)^{16} = ( (1+i)^{12} \times (1+i)^4 ) \]\[ = -64 \times (-4) = 256 \].

Key Concepts

Binomial Theorem

The binomial theorem is a powerful tool in algebra that allows us to expand expressions that are raised to a power, specifically binomials like \((a + b)^n\).Using the binomial theorem, you can generate the expanded form using binomial coefficients. These coefficients are determined from Pascal’s triangle, and they guide you in forming each term of the expanded expression.Let's break it down how it's used with the expression \((1 + i)^4\):

• The binomial theorem expression for \((a + b)^n\) is: \[(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\]
• Here, \(a = 1\) and \(b = i\).For \((1 + i)^4\), you will get terms like \(\binom{4}{0} \cdot 1^4 \cdot i^0\), which simplifies to \(1\).
• Similarly, for each term, compute the powers and combine them using the coefficients: \(\binom{4}{1} \cdot 1^3 \cdot i\), \(\binom{4}{2} \cdot 1^2 \cdot i^2\), and so on.

After computing, you find the expression simplifies via addition of terms, where some results, like \(i^2 = -1\), change the operation. Understanding the binomial theorem helps in simplifying calculations significantly compared to directly multiplying the components.

Powers of Complex Numbers

Calculating powers of complex numbers like \((1+i)\) involves both algebraic manipulation and pattern recognition.Complex numbers are numbers of the form \(a + bi\), where \(i\) is the imaginary unit with the property \(i^2 = -1\). To find powers of complex numbers, one effective method is using the known powers of simpler combinations like \((1 + i)\).For example:

• In Step 1, \((1+i)^4\) simplifies to \(-4\). It’s achieved by using the binomial theorem to reduce the expansion.
• In Step 2, once \((1+i)^4\) is known, determine \((1+i)^8\) by squaring: \((-4)^2 = 16\).
• In Step 3, \((1+i)^{12}\) is determined by cubing the result of Step 1: \((-4)^3 = -64\).

These examples show how calculated powers provide a guide to get higher powers, and by using results from smaller power computations, much larger powers become manageable. This saves time and avoids complex multiplications.

Patterns in Mathematics

Mathematical patterns can simplify complex problems and help predict results without performing complete calculations. Recognizing patterns is invaluable when dealing with complex numbers.By observing how \((1+i)\) behaves when raised to increasing powers, a pattern emerges:

• For \((1+i)^4\), it simplifies to \(-4\).
• In \((1+i)^8\), it becomes \(16\), multiplying the result of \((1+i)^4\) by itself.
• \((1+i)^{12}\) transitions to \(-64\), indicating multiplication by \(-4\) again.
• Thus, predicting \((1+i)^{16}\) involves continuing the pattern, expecting \(256\) as it multiplies by \(-4\) once more.

Patterns simplify the guessing of the next number in sequences. They offer shortcuts for complex numbers and higher power calculations, improving efficiency. When you identify a pattern, predictions become logical guesses rather than trial and error computations.