Problem 92 - Intermediate Algebra

Question

Find each power of \(i .\) See Example 6. $$ (-2 i)^{7} $$

Step-by-Step Solution

Answer

The result is \(128i\).

Step 1: Express the Power as a Product

First, we write the expression \((-2i)^7\) as \((-2)^7 imes i^7\). This separates the constant from the power of \(i\).

Step 2: Calculate the Power of the Constant

Let's calculate \((-2)^7\):\((-2)^7 = -2 \times -2 \times -2 \times -2 \times -2 \times -2 \times -2\).Applying multiplication, we find \((-2)^7 = -128\).

Step 3: Determine the Power of i

Remember, the powers of \(i\) cycle in a pattern: - \(i^1 = i\)- \(i^2 = -1\)- \(i^3 = -i\)- \(i^4 = 1\). Since 7 mod 4 equals 3, we find that \(i^7 = i^3 = -i\).

Step 4: Combine the Results

Now multiply the results obtained from Step 2 and Step 3:The value is \((-128) imes (-i) = 128i\).

Key Concepts

Powers of Imaginary Unit

When dealing with complex numbers, especially those including the imaginary unit, denoted as \( i \), it’s crucial to understand the cyclical nature of its powers. The imaginary unit \( i \) is mathematically significant because \( i^2 = -1 \). This property sets off a repeating cycle every four powers:

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

Each fourth power brings us back to \(1\), repeating this sequence for higher powers of \( i \).
This means that you only need to determine a power's remainder when divided by 4 to know its equivalence. For example, if you are finding \( i^7 \), simply calculate the remainder of 7 divided by 4, which equals 3. Therefore, \( i^7 = i^3 = -i \). This simplicity turns potentially complex calculations into manageable steps.

Integer Exponents

In mathematics, dealing with integer exponents is a fundamental skill that helps address various expressions, including those involving complex numbers. When raising a number to an integer exponent, you're essentially multiplying that number by itself as many times as the value of the exponent.
For instance, consider \((-2)^7\). This implies multiplying \(-2\) by itself seven times:

• \(-2 \times -2 \times -2 \times -2 \times -2 \times -2 \times -2 \)

By calculating this step-by-step:

• \((-2 \times -2) = 4 \)
• \(4 \times -2 = -8 \)
• \(-8 \times -2 = 16 \), and so on until you reach \(-128\).

Exponentiation is useful in simplifying expressions, helping both in calculations and in understanding the behaviors of more complex functions. Remember that exponents apply to constants and imaginary numbers alike, so the process remains consistent.

Cyclic Patterns in Math

Cyclic patterns are a recurring theme throughout mathematics and are especially prevalent when working with powers, like those of the imaginary unit \( i \). Essentially, any sequence that repeats itself after a set number of terms can be considered a cyclic pattern.
The powers of \( i \) (\( i^1, i^2, i^3, i^4 \) and their repetitions) serve as a perfect example, repeating every four terms:

• This repetition simplifies many complex calculations, allowing mathematicians to predict and verify outcomes efficiently.

Recognizing cyclic patterns aids in problem-solving by allowing one to forecast results without performing repetitive multiplication.
This is particularly useful in calculating large powers. For instance, if tasked to find \( i^{100}\), you can simply find the remainder of 100 divided by 4, which is 0, corresponding to \( i^4 = 1 \).
These patterns not only simplify computations but also deepen understanding of the structure within mathematical systems, leading to more intuitive reasoning and problem-solving.