The 5 values of (1+i)^3 / 5 are given by 2^3 / 10[cos((3 π)/(20)+(6 k π)/(5))+i sin((3 π)/(20)+(

step 1 In this question there are two statements and based on the statements we need to find which opti

The 5 values of (1+i)^3 / 5 are given by 2^3 / 10[cos((3...

The 5 values of $(1+\mathrm{i})^{3 / 5}$ are given by $2^{3 / 10}\left[\cos \left(\frac{3 \pi}{20}+\frac{6 \mathrm{k} \pi}{5}\right)+\mathrm{i} \sin \left(\frac{3 \pi}{20}+\frac{6 \mathrm{k} \pi}{5}\right)\right]$, where $\mathrm{k}=0,1,2,3,4$ and Statement 2 If $z=r(\cos \alpha+i \sin \alpha)$ and $n$ is a positive integer, the $n$ values of $z^{1 / n}$ are given by $r^{1 / n}\left(\cos \left(\frac{\alpha}{n}+\frac{2 k \pi}{n}\right)+i \sin \left(\frac{\alpha}{n}+\frac{2 k \pi}{n}\right)\right.$ $\mathrm{k}=0,1,2, \ldots \ldots, \mathrm{n}-1$

The 5 values of $(1+\mathrm{i})^{3 / 5}$ are given by $2^{3 / 10}\left[\cos \left(\frac{3 \pi}{20}+\frac{6 \mathrm{k} \pi}{5}\right)+\mathrm{i} \sin \left(\frac{3 \pi}{20}+\frac{6 \mathrm{k} \pi}{5}\right)\right]$, where $\mathrm{k}=0,1,2,3,4$
and
Statement 2 If $z=r(\cos \alpha+i \sin \alpha)$ and $n$ is a positive integer, the $n$ values of $z^{1 / n}$ are given by $r^{1 / n}\left(\cos \left(\frac{\alpha}{n}+\frac{2 k \pi}{n}\right)+i \sin \left(\frac{\alpha}{n}+\frac{2 k \pi}{n}\right)\right.$
$\mathrm{k}=0,1,2, \ldots \ldots, \mathrm{n}-1$

Key Concepts

De Moivre's Theorem

De Moivre's theorem states that for a complex number expressed in polar form, z = r(cos ? + i sin ?), its power can be calculated as z^n = r^n (cos n? + i sin n?). The theorem also underpins root extraction by allowing one to distribute the exponent over the modulus and the angle, leading to expressions for the roots.

Nth Roots of Complex Numbers

Extracting the nth roots of a complex number involves finding all possible values of z such that when raised to the power n, the result is the original complex number. The formula r^(1/n)(cos((? + 2k?)/n) + i sin((? + 2k?)/n)), with k = 0, 1, ..., n – 1, captures all distinct solutions by accounting for the periodicity of the trigonometric functions.

Complex Numbers in Polar Form

This concept involves expressing a complex number in the form r(cos ? + i sin ?), where r is the modulus (or absolute value) and ? is the argument (or angle). This representation simplifies multiplication, division, and finding powers or roots of complex numbers by translating operations into modifications of r and ?.

Equidistant Angles on the Complex Plane

When computing the nth roots of a complex number, the resulting values have arguments that are evenly spaced around a circle. The term 2k?/n in the root formula ensures that the roots are spread uniformly, forming vertices of a regular polygon inscribed in a circle with radius r^(1/n).

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