Simplifying I^15: A Step-by-Step Guide

Hey guys! Let's dive into the fascinating world of imaginary numbers and tackle a common problem: simplifying $i^{15}$. This might seem daunting at first, but with a few key concepts, it becomes super easy. So, grab your thinking caps, and let's get started!

Understanding the Imaginary Unit i

Before we jump into simplifying $i^{15}$, it's crucial to understand what i actually represents. The imaginary unit i is defined as the square root of -1. Mathematically, we write it as $i = \sqrt{-1}$. This is the foundation of complex numbers, which extend the real number system by including this imaginary unit. Now, you might be wondering, why do we even need such a thing? Well, imaginary numbers allow us to solve equations that have no real solutions, like $x^2 + 1 = 0$. By introducing i, we can say that the solutions to this equation are $x = i$ and $x = -i$.

The powers of i follow a cyclical pattern, which is key to simplifying higher powers like $i^{15}$. Let's explore this pattern in detail:

• $i^1 = i$
• $i^2 = -1$ (since $i = \sqrt{-1}$, then $i^2 = (\sqrt{-1})^2 = -1$)
• $i^3 = i^2 * i = -1 * i = -i$
• $i^4 = i^2 * i^2 = (-1) * (-1) = 1$

And here's where the magic happens: the pattern repeats! This means $i^5 = i$, $i^6 = -1$, $i^7 = -i$, $i^8 = 1$, and so on. This cyclical nature is what allows us to simplify any power of i by finding its remainder when divided by 4. This is because every four powers, the pattern repeats itself. Knowing this pattern is fundamental to simplifying $i^{15}$. We will leverage this to break down the exponent and find the equivalent value within the cycle.

Understanding this cycle not only helps in simplifying powers of i but also provides a deeper understanding of complex numbers and their applications in various fields like electrical engineering, quantum mechanics, and signal processing. So, keep this pattern in mind as we move forward and tackle more complex problems involving imaginary numbers. With this knowledge, you'll be able to simplify any power of i with ease and confidence!

Simplifying $i^{15}$

Now that we've got a solid grasp of what i is and the cyclic pattern of its powers, let's get to the main event: simplifying $i^{15}$. The trick here is to use the fact that $i^4 = 1$. We want to break down $i^{15}$ into a product involving $i^4$ raised to some power, and then whatever is left over. To do this, we divide the exponent (15) by 4.

When we divide 15 by 4, we get 3 with a remainder of 3. This can be written as $15 = 4 * 3 + 3$. So, we can rewrite $i^{15}$ as follows:

$i^{15} = i^{(4*3 + 3)}$

Using the properties of exponents, we can further break this down:

$i^{(4*3 + 3)} = i^{(4*3)} * i^3 = (i^4)^3 * i^3$

Since we know that $i^4 = 1$, we can substitute that in:

$(i^4)^3 * i^3 = (1)^3 * i^3 = 1 * i^3 = i^3$

So, now we have $i^{15} = i^3$. Remember from our earlier discussion that $i^3 = -i$. Therefore:

$i^{15} = -i$

And that's it! We've successfully simplified $i^{15}$ to $-i$. The key was recognizing the cyclical pattern of powers of i and using the fact that $i^4 = 1$ to break down the exponent. By dividing the exponent by 4 and looking at the remainder, we can quickly determine the simplified value of any power of i. This approach makes simplifying higher powers of i straightforward and manageable. Practice with different exponents to master this technique and become a pro at simplifying imaginary numbers!

Examples of Simplifying Other Powers of i

To solidify your understanding, let's walk through a couple more examples of simplifying different powers of i. This will help you see how the same principles apply to various exponents and reinforce the technique we just used. By practicing these examples, you'll become more confident and proficient in simplifying any power of i. Remember, the key is to divide the exponent by 4 and focus on the remainder.

Example 1: Simplify $i^{22}$

1. Divide the exponent by 4: $22 ÷ 4 = 5$ with a remainder of $2$.
2. Rewrite $i^{22}$ using the remainder: $i^{22} = i^{(4*5 + 2)} = (i^4)^5 * i^2$.
3. Simplify using $i^4 = 1$: $(i^4)^5 * i^2 = (1)^5 * i^2 = 1 * i^2 = i^2$.
4. Since $i^2 = -1$, we have $i^{22} = -1$.

So, $i^{22}$ simplifies to $-1$.

Example 2: Simplify $i^{37}$

1. Divide the exponent by 4: $37 ÷ 4 = 9$ with a remainder of $1$.
2. Rewrite $i^{37}$ using the remainder: $i^{37} = i^{(4*9 + 1)} = (i^4)^9 * i^1$.
3. Simplify using $i^4 = 1$: $(i^4)^9 * i^1 = (1)^9 * i^1 = 1 * i = i$.
4. Therefore, $i^{37} = i$.

So, $i^{37}$ simplifies to i. These examples demonstrate how consistently applying the same method allows you to simplify any power of i. By breaking down the exponent and using the cyclical property, you can easily find the simplified value without having to memorize a huge list of powers. The most important concept to remember is the cyclical pattern of i and how it relates to division by 4. Keep practicing, and you'll become a master of simplifying imaginary numbers!

Conclusion

Alright, guys, we've covered a lot! Simplifying $i^{15}$ might have seemed tricky at first, but by understanding the basic definition of i and the cyclical pattern of its powers, we were able to break it down into manageable steps. Remember the key takeaway: divide the exponent by 4 and focus on the remainder. This remainder will tell you which value in the cycle ($i, -1, -i, 1$) corresponds to the simplified form of the power of i. This method isn't just useful for $i^{15}$; it works for any power of i! The examples we worked through should give you the confidence to tackle any similar problem.

Complex numbers, with their imaginary components, are a fundamental part of many areas of mathematics, physics, and engineering. Mastering the simplification of powers of i is a stepping stone to understanding more advanced concepts in these fields. So keep practicing, keep exploring, and don't be afraid to dive deeper into the fascinating world of complex numbers. Who knows what other mathematical wonders you'll discover! By understanding the behavior of i, you unlock a deeper understanding of complex numbers and their applications. So go forth and simplify, and have fun doing it! And remember, math isn't scary; it's just a puzzle waiting to be solved. Keep practicing, and you'll become a pro in no time!Remember that practice makes perfect, so keep working on problems, and soon you'll be simplifying powers of i like a pro!