Exercise · ZW-COMPLEX-NTH-ROOT-007 · ★★☆ Intermediate
Find the square roots of \(i\).
Choose one option.
Show answer
- C. \(\pm\frac{\sqrt2}{2}(1+i)\)
Explanation
\(r=1\), \(\theta=\frac\pi2\): \(\cos\frac\pi4+i\sin\frac\pi4=\frac{\sqrt2}2(1+i)\) and its negative. Check: \(\left(\frac{\sqrt2}2(1+i)\right)^2=\frac12\cdot2i=i\).
Review the concept: Complex nth root
An \(n\)-th root of a complex number \(w\) is a complex number \(z\) whose \(n\)-th power, with \(n\) a positive integer, is \(w\). If \(w\) has modulus \(r\) and argument \(\theta\), its roots are given below.
\[z_k=\sqrt[n]{r}\left(\cos\frac{\theta+2\pi k}{n}+i\sin\frac{\theta+2\pi k}{n}\right),\quad k=0,\dots,n-1\]
More practice problems: complex nth root
- ZW-COMPLEX-NTH-ROOT-001 How many distinct cube roots does 8 have in ℂ?
- ZW-COMPLEX-NTH-ROOT-002 Find the square roots of -4.
- ZW-COMPLEX-NTH-ROOT-003 Which of these is NOT a fourth root of 1?
- ZW-COMPLEX-NTH-ROOT-004 What is the modulus of each fifth root of 32?
- ZW-COMPLEX-NTH-ROOT-005 The cube root of -8 with k=0 in the formula is:
- ZW-COMPLEX-NTH-ROOT-006 The sum of the n n-th roots of 1, with n≥2, is:
- ZW-COMPLEX-NTH-ROOT-008 The angle between two consecutive sixth roots of any nonzero number is:
- ZW-COMPLEX-NTH-ROOT-009 Which of these is a cube root of -27i?