# Two formulas for calculating root of a complex number in a exponential form

• R A V E N
In summary, the correct version is the first one where the entire term \left(re^{i(\phi+2k\pi)}\right) is under the root, while the second one is incorrect because it only has re under the root.
R A V E N
$$z_k=\sqrt[n]{u}=\sqrt[n]{r}e^{i\left(\frac{\phi+2k\pi}{n}\right)}, k=0,1,2,...,n-1$$

and

$$z_k=\sqrt[n]{u}=\sqrt[n]{re}^\frac{\phi+2k\pi}{n}, k=0,1,2,...,n-1$$

Which one is incorrect (note that in the first, $$e$$ is out of the root)?

The second one is incorrect, and I would read that one as being $$\left(\sqrt[n]{re}\right)^\frac{\phi+2k\pi}{n}$$

When you have $$z_k^n=re^{i(\phi+2k\pi)}$$

then taking the nth would yield $$z_k=\left(re^{i(\phi+2k\pi)}\right)^{\frac{1}{n}}=r^{\frac{1}{n}}e^{i(\phi+2k\pi)\frac{1}{n}}=\sqrt[n]{r}e^{i(\frac{\phi+2k\pi}{n})}$$

## What is a complex number in exponential form?

A complex number in exponential form is a number that can be expressed as a combination of a real number and an imaginary number multiplied by the base of the natural logarithm, e. It is written in the form re^(iθ), where r is the magnitude or modulus of the complex number and θ is the angle or argument of the complex number.

## What is the first formula for calculating the root of a complex number in exponential form?

The first formula for calculating the root of a complex number in exponential form is the polar form of De Moivre's theorem. It states that for a complex number z = re^(iθ), the nth root can be calculated as r^(1/n)e^(iθ/n), where n is the desired root.

## What is the second formula for calculating the root of a complex number in exponential form?

The second formula for calculating the root of a complex number in exponential form is the exponential form of the nth root formula. It states that for a complex number z = re^(iθ), the nth root can be calculated as (r^(1/n))e^(i(θ + 2kπ)/n), where k is an integer representing the different roots.

## What is the difference between the two formulas for calculating the root of a complex number in exponential form?

The first formula (polar form of De Moivre's theorem) uses the modulus and argument of the complex number to calculate the root, while the second formula (exponential form of the nth root formula) uses the modulus and angle of the complex number, taking into account the different roots by adding 2kπ to the angle.

## Can these formulas be used for all types of complex numbers?

Yes, these formulas can be used for all types of complex numbers, including pure imaginary numbers and real numbers. However, for real numbers, the angle θ is equal to 0 and the formulas simplify to r^(1/n) for the first formula and r^(1/n)e^(2kπ/n) for the second formula.

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