# De Moivre’s Theorem Calculator

Need to calculate powers of complex numbers? The **De Moivre’s Theorem Calculator** is here to help. Enter the complex number, exponent, magnitude and argument required values into the calculator to find the complex number raised to a given power.

### What is De Moivre’s Theorem?

De Moivre’s Theorem is a formula that links complex numbers and trigonometry. It allows you to raise a complex number to any power using polar coordinates.

### How to Use the Calculator

To use the De Moivre’s Theorem Calculator is straightforward. Here’s how:

**Complex Number (z)**: Enter the complex number. For example, type`2 + 3i`

.**Exponent (n)**: Enter the exponent. For instance, type`4`

.**Magnitude (r)**: Enter the magnitude. Try`3.6`

.**Argument (θ)**: Enter the argument in radians. Enter`0.927`

.

Click “Calculate” to get the result.

### Formula

The formula for De Moivre’s Theorem is:

Variables

Variable | Description |
---|---|

$z$ | Complex number |

$n$ | Exponent |

$r$ | Magnitude of the complex number |

$\theta $ | Argument (angle) in radians |

### Calculation Examples

#### Example 1

**Complex Number**: $2+3i$**Exponent**: $4$**Magnitude**: $3.6$**Argument**: $0.927$

**Step-by-Step Calculation**:

- Identify the values: $z=2+3i$, $n=4$, $r=3.6$, $\theta =0.927$
- Calculate the new magnitude: ${r}^{n}=3.{6}^{4}=167.9616$.
- Calculate the new argument: $n\theta =4\times 0.927=3.708$.
- Find the real part: $167.9616\cdot \mathrm{cos}(3.708)=-147.310$.
- Find the imaginary part: $167.9616\cdot \mathrm{sin}(3.708)=-83.983$.

**Result**: $-147.310+-83.983i$.

#### Example 2

**Complex Number**: $1+i$**Exponent**: $3$**Magnitude**: $1.414$**Argument**: $0.785$

**Step-by-Step Calculation**:

- Identify the values: $z=1+i$, $n=3$, $r=1.414$, $\theta =0.785$.
- Calculate the new magnitude: ${r}^{n}=1.41{4}^{3}=2.828$.
- Calculate the new argument: $n\theta =3\times 0.785=2.355$.
- Find the real part: $2.828\cdot \mathrm{cos}(2.355)=-2.0$.
- Find the imaginary part: $2.828\cdot \mathrm{sin}(2.355)=2.828$.

**Result**: $-2.0+2.828i$.

I hope by using our De Moivre’s Theorem Calculator, you did effortlessly raised any complex number to a given power. Please lets us know on report error page if you are facing any issue while using the tool.