The de Moivre theorem explains how to calculate the powers of complex numbers.

For any complex number x and any integer n,

- Enter the complex number z into the calculator. Complex numbers are typically written in the form
**a + b**, where a is the real part and b is the imaginary part. For example, the complex number*i***3 + 4**would be entered as*i***3+4**.*i* - Enter the positive integer
**n**into the calculator. This is the power to which you want to raise the complex number**z**. - Click the
**"Calculate"**button to compute**(z**based on De Moivre's Theorem.^{n}) - The calculator will display the result in both polar and rectangular form. The polar form of a complex number is written as
*(r, θ)*, where**r**is the modulus*(magnitude)*of the number and*θ*is the argument*(angle)*in polar coordinates. The rectangular form of a complex number is written as**a + b**, where*i***a**is the real part and**b**is the imaginary part.

For example, if you entered the complex number **3 + 4 i** and the integer 3 into the calculator, the result would be displayed as

De Moivre's Theorem is a useful mathematical result that has a number of applications in various fields. Some of the key uses of De Moivre's Theorem include:

**Complex number manipulation:**De Moivre's Theorem allows us to easily perform calculations involving complex numbers, such as raising them to a power or finding their roots.**Trigonometry:**De Moivre's Theorem can be used to derive trigonometric identities and to simplify complex trigonometric expressions.**Eigenvalues and eigenvectors:**In linear algebra, De Moivre's Theorem is used to find the eigenvalues and eigenvectors of matrices.**Signal processing:**De Moivre's Theorem is used in the analysis and processing of signals, such as audio and image signals.**Quantum mechanics:**De Moivre's Theorem is used in the study of quantum mechanics to describe the behavior of particles and systems at the atomic and subatomic level.**Other fields:**De Moivre's Theorem has applications in various other fields, including engineering, computer science, and economics.