Problem 38 - Algebra 2 and Trigonometry

Question

In \(35-43,\) write each number in simplest form. $$ 2 i^{5}+7 i^{7} $$

Step-by-Step Solution

Answer

The simplified form of the expression is \(-5i\).

Step 1: Understand the Power of i

The imaginary unit, represented by \(i\), is defined as the square root of -1. A key property of \(i\) is that it has a cyclical power pattern. For example: \(i^1=i\); \(i^2=-1\); \(i^3=-i\); \(i^4=1\); and this cycle repeats. Recognize and memorize these basic properties: \(i^5 = i\), \(i^6 = -1\), \(i^7 = -i\), \(i^8 = 1\), and so on.

Step 2: Simplify Each Term Separately

Given the expression \(2i^5 + 7i^7\). Using the cyclic nature of \(i\) powers, simplify each term: \(i^5 = i\) and \(i^7 = -i\). Thus, the expression becomes \(2i + 7(-i)\).

Step 3: Combine Like Terms

Now that we have \(2i\) and \(-7i\), combine the like terms. Calculate \(2i - 7i = -5i\).

Step 4: Final Simplified Form

The simplified form of the expression \(2i^5 + 7i^7\) is \(-5i\).

Key Concepts

Imaginary Unit

The imaginary unit, commonly denoted as \(i\), is a fundamental concept in complex numbers. It's defined as \(i=\sqrt{-1}\). This definition is crucial because no real number squared gives a negative number, so \(i\) allows us to work with square roots of negative numbers.
To comprehend \(i\), remember these key points:

• \(i\) is not a real number but is the basis of imaginary numbers.
• \(i^2 = -1\), which is the defining property that helps simplify complex expressions.

Understanding \(i\) transforms how we handle negative square roots and allows us to explore solutions outside the realm of conventional mathematics.

Powers of i

A fascinating aspect of \(i\) is its cyclic pattern with powers. Whenever you raise \(i\) to higher powers, it cycles through four main results repeatedly. Recognizing this pattern is vital for simplifying expressions involving \(i\).
The basic cycle for \(i\) is:

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

This cycle repeats, meaning:

• \(i^5 = i\)
• \(i^6 = -1\)
• and so forth...

When simplifying powers of \(i\), identify the power's remainder when divided by 4. This remainder links to one of the four recurring outcomes above, allowing you to simplify complex expressions easily.

Simplifying Expressions

When confronted with complex expressions involving powers of \(i\), the goal is to break down each component using the cyclical properties of \(i\). Let's look at how this works using the example: \(2i^5 + 7i^7\).
Start by using what you know about the pattern of \(i\):

• \(i^5\) simplifies to \(i\) because it's the same as \(i^1\), since 5 mod 4 leaves a remainder of 1.
• \(i^7\) simplifies to \(-i\), as it corresponds to \(i^3\), with 7 mod 4 leaving a remainder of 3.

Now plug these values back into the original expression:

• Convert \(2i^5\) to \(2i\).
• Change \(7i^7\) to \(7(-i) = -7i\).

You now have \(2i - 7i\).
Combine these like terms to get the final simplified form: \(-5i\).
Properly simplifying expressions involving \(i\) requires a thorough understanding of its powers' cyclicity, letting you break down and accurately solve more complex problems.