Find ( 2 ( Cos ⁡ 2 Π 3 + I Sin ⁡ 2 Π 3 ) ) 5 \left(2\left(\cos \frac{2 \pi}{3}+i \sin \frac{2 \pi}{3}\right)\right)^5 ( 2 ( Cos 3 2 Π ​ + I Sin 3 2 Π ​ ) ) 5 .A. − 16 3 − 16 I -16 \sqrt{3}-16 I − 16 3 ​ − 16 I B. − 16 − 16 I 3 -16-16 I \sqrt{3} − 16 − 16 I 3 ​ C. 16 + 16 I 3 16+16 I \sqrt{3} 16 + 16 I 3 ​ D. 16 3 + 16 I 16 \sqrt{3}+16 I 16 3 ​ + 16 I

Introduction

De Moivre's Theorem is a powerful tool in mathematics that allows us to find the value of complex numbers raised to a power. This theorem is named after the French mathematician Abraham de Moivre, who first discovered it in the 18th century. In this article, we will explore how to use De Moivre's Theorem to find the value of a complex number raised to a power, specifically the expression $\left(2\left(\cos \frac{2 \pi}{3}+i \sin \frac{2 \pi}{3}\right)\right)^5$.

Understanding Complex Numbers

Before we dive into De Moivre's Theorem, let's take a moment to understand complex numbers. A complex number is a number that can be expressed in the form $a+bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit, which satisfies the equation $i^2=-1$. The real part of a complex number is $a$, and the imaginary part is $b$. Complex numbers can be represented graphically on a complex plane, with the real part on the x-axis and the imaginary part on the y-axis.

Polar Form of Complex Numbers

To apply De Moivre's Theorem, we need to express the complex number in polar form. The polar form of a complex number is given by $r(\cos \theta + i \sin \theta)$, where $r$ is the magnitude of the complex number and $\theta$ is the angle between the positive x-axis and the line segment connecting the origin to the complex number. In the given expression, we have $2\left(\cos \frac{2 \pi}{3}+i \sin \frac{2 \pi}{3}\right)$, which is already in polar form.

De Moivre's Theorem

De Moivre's Theorem states that for any complex number $z=r(\cos \theta + i \sin \theta)$ and any integer $n$, we have $z^n=r^n(\cos n\theta + i \sin n\theta)$. This theorem allows us to find the value of a complex number raised to a power by simply raising the magnitude to the power and multiplying the angle by the power.

Applying De Moivre's Theorem

Now that we have De Moivre's Theorem, let's apply it to the given expression. We have $\left(2\left(\cos \frac{2 \pi}{3}+i \sin \frac{2 \pi}{3}\right)\right)^5$. Using De Moivre's Theorem, we can rewrite this expression as $2^5\left(\cos 5\left(\frac{2 \pi}{3}\right) + i \sin 5\left(\frac{2 \pi}{3}\right)\right)$.

Evaluating the Expression

Now that we have the expression in the form $r^n(\cos n\theta + i \sin n\theta)$, we can evaluate it. We have $2^5=32$, and we need to find the values of $\cos 5\left(\frac{2 \pi}{3}\right)$ and $\sin 5\left(\frac{2 \pi}{3}\right)$. Using the unit circle, we can find that $\cos 5\left(\frac{2 \pi}{3}\right)=-\frac{\sqrt{3}}{2}$ and $\sin 5\left(\frac{2 \pi}{3}\right)=-\frac{1}{2}$.

Simplifying the Expression

Now that we have the values of $\cos 5\left(\frac{2 \pi}{3}\right)$ and $\sin 5\left(\frac{2 \pi}{3}\right)$, we can simplify the expression. We have $32\left(-\frac{\sqrt{3}}{2}-\frac{1}{2}i\right)$. Multiplying the terms, we get $-16\sqrt{3}-16i$.

Conclusion

In this article, we used De Moivre's Theorem to find the value of the complex number $\left(2\left(\cos \frac{2 \pi}{3}+i \sin \frac{2 \pi}{3}\right)\right)^5$. We first expressed the complex number in polar form, then applied De Moivre's Theorem to find the value of the complex number raised to a power. Finally, we simplified the expression to find the final answer.

Answer

The final answer is $\boxed{-16\sqrt{3}-16i}$.

Discussion

This problem is a great example of how De Moivre's Theorem can be used to find the value of complex numbers raised to a power. The theorem is a powerful tool in mathematics that allows us to simplify complex expressions and find the values of complex numbers. In this problem, we used the theorem to find the value of a complex number raised to a power, and we were able to simplify the expression to find the final answer.

Related Problems

If you are interested in learning more about De Moivre's Theorem and how to use it to find the value of complex numbers raised to a power, you may want to try the following problems:

• Find the value of $\left(3\left(\cos \frac{\pi}{4}+i \sin \frac{\pi}{4}\right)\right)^4$
• Find the value of $\left(2\left(\cos \frac{3 \pi}{4}+i \sin \frac{3 \pi}{4}\right)\right)^3$
• Find the value of $\left(4\left(\cos \frac{\pi}{3}+i \sin \frac{\pi}{3}\right)\right)^2$

These problems are great examples of how De Moivre's Theorem can be used to find the value of complex numbers raised to a power, and they are a great way to practice using the theorem.

Introduction

In our previous article, we explored De Moivre's Theorem and how to use it to find the value of complex numbers raised to a power. In this article, we will answer some common questions about De Moivre's Theorem and provide additional examples to help you practice using the theorem.

Q: What is De Moivre's Theorem?

A: De Moivre's Theorem is a mathematical formula that allows us to find the value of complex numbers raised to a power. It states that for any complex number $z=r(\cos \theta + i \sin \theta)$ and any integer $n$, we have $z^n=r^n(\cos n\theta + i \sin n\theta)$.

Q: How do I apply De Moivre's Theorem?

A: To apply De Moivre's Theorem, you need to follow these steps:

1. Express the complex number in polar form.
2. Raise the magnitude to the power.
3. Multiply the angle by the power.
4. Simplify the expression.

Q: What are some common mistakes to avoid when using De Moivre's Theorem?

A: Here are some common mistakes to avoid when using De Moivre's Theorem:

• Not expressing the complex number in polar form.
• Not raising the magnitude to the power.
• Not multiplying the angle by the power.
• Not simplifying the expression.

Q: Can I use De Moivre's Theorem to find the value of complex numbers raised to a fractional power?

A: No, De Moivre's Theorem only works for integer powers. If you need to find the value of a complex number raised to a fractional power, you will need to use a different method.

Q: How do I find the value of a complex number raised to a negative power?

A: To find the value of a complex number raised to a negative power, you can use De Moivre's Theorem with a negative exponent. For example, if you need to find the value of $z^{-n}$, you can use the formula $z^{-n}=\frac{1}{r^n(\cos n\theta + i \sin n\theta)}$.

Q: Can I use De Moivre's Theorem to find the value of complex numbers raised to a power in the form $a+bi$?

A: No, De Moivre's Theorem only works for complex numbers in polar form. If you need to find the value of a complex number raised to a power in the form $a+bi$, you will need to convert the complex number to polar form first.

Q: How do I find the value of a complex number raised to a power using De Moivre's Theorem when the angle is not a multiple of $\pi$?

A: To find the value of a complex number raised to a power using De Moivre's Theorem when the angle is not a multiple of $\pi$, you can use the formula $z^n=r^n(\cos n\theta + i \sin n\theta)$, where $\theta$ is the angle in radians.

Q: Can I use De Moivre's Theorem to find the value of complex numbers raised to a power in the form $r(\cos \theta + i \sin \theta)$?

A: Yes, De Moivre's Theorem works for complex numbers in the form $r(\cos \theta + i \sin \theta)$. You can simply raise the magnitude to the power and multiply the angle by the power.

Conclusion

In this article, we answered some common questions about De Moivre's Theorem and provided additional examples to help you practice using the theorem. We hope this article has been helpful in understanding De Moivre's Theorem and how to use it to find the value of complex numbers raised to a power.

Practice Problems

Here are some practice problems to help you practice using De Moivre's Theorem:

• Find the value of $\left(3\left(\cos \frac{\pi}{4}+i \sin \frac{\pi}{4}\right)\right)^4$
• Find the value of $\left(2\left(\cos \frac{3 \pi}{4}+i \sin \frac{3 \pi}{4}\right)\right)^3$
• Find the value of $\left(4\left(\cos \frac{\pi}{3}+i \sin \frac{\pi}{3}\right)\right)^2$

Additional Resources

If you are interested in learning more about De Moivre's Theorem and how to use it to find the value of complex numbers raised to a power, you may want to check out the following resources:

• Khan Academy: De Moivre's Theorem
• Mathway: De Moivre's Theorem
• Wolfram Alpha: De Moivre's Theorem

These resources provide additional examples and explanations to help you understand De Moivre's Theorem and how to use it to find the value of complex numbers raised to a power.