Problem 78 - Algebra A Combined Function

Question

Find each power of \(i\). $$ i^{-9} $$

Step-by-Step Solution

Answer

\(i^{-9} = -i\).

Step 1: Understand the Power of i

The imaginary unit \(i\) is defined such that \(i^2 = -1\). This property leads to a cycle for powers of \(i\): \(i^1 = i\), \(i^2 = -1\), \(i^3 = -i\), and \(i^4 = 1\). This cycle repeats every four exponents, so the power of \(i\) depends on the remainder when the exponent is divided by 4.

Step 2: Determine the Equivalent Positive Power

To evaluate \(i^{-9}\), first convert the negative exponent to a positive exponent using the identity \(i^{-n} = \frac{1}{i^n}\). We need to find \(i^9\) instead, and then take the reciprocal.

Step 3: Find the Remainder Using Modulo

Since the powers of \(i\) complete a cycle every 4 terms, compute \(9 \mod 4\) to find the equivalent power within the cycle. This gives \(9 \div 4 = 2\) remainder 1, so \(i^9\) is equivalent to \(i^1\).

Step 4: Reciprocal of the Power

Now that we know \(i^9 = i^1 = i\), \(i^{-9}\) is \(\frac{1}{i}\). Simplify \(\frac{1}{i}\) by multiplying the numerator and denominator by \(-i\) to get \(-i\). Hence, \(i^{-9} = -i\).

Key Concepts

Imaginary Unit

The imaginary unit, denoted by \(i\), serves as the foundation for complex numbers. It's defined as the square root of -1. A tricky concept to grasp at first, but it allows us to explore the realm of numbers that extend beyond the real number line.

Key points to remember about \(i\):

• \(i^2 = -1\)
• Used primarily in defining complex numbers, which take the form \(a + bi\) where \(a\) and \(b\) are real numbers.
• Enables mathematical operations like addition, multiplication, and exponentiation within the complex plane.

This understanding of \(i\) allows it to play a pivotal role in advanced mathematics, physics, engineering, and various technology fields.

Powers of i

The powers of \(i\) display a repetitive cycle every four exponents, making it easier to calculate large powers. This cycle consists of four key values:

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

After \(i^4\), these values repeat for subsequent powers. To find \(i^9\), calculate the remainder of 9 divided by 4, which is 1. Hence, \(i^9 = i^1 = i\).

Using this cyclical pattern helps simplify seemingly complex calculations with ease, saving time and reducing computational errors. It demonstrates the elegance and consistency of mathematical structures.

Negative Exponents

Negative exponents often make calculations appear daunting, but they can be easily simplified. The concept involves turning a negative exponent into a positive one by taking the reciprocal of the base number raised to the positive equivalent exponent.

In the context of our example \(i^{-9}\):

• Convert to positive exponent: \(i^{-9} = \frac{1}{i^9}\).
• Previously, we found \(i^9 = i\), so \(\frac{1}{i} = -i\).

This conversion turns potential complexity into simplicity by employing reciprocal operations. It eases problem-solving for exponential problems, especially in the realms of algebra and higher mathematics.