Exercise · ZW-COMPLEX-NTH-ROOT-002 · ★☆☆ Basic
Find the square roots of \(-4\).
Choose one option.
Show answer
- D. \(2i\) and \(-2i\)
Explanation
\((\pm2i)^2=-4\). For \(w=-4\): \(r=4\), \(\theta=\pi\), and \(\sqrt4\,(\cos\frac\pi2+i\sin\frac\pi2)=2i\) plus its negative.
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
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- 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-007 Find the square roots of i.
- ZW-COMPLEX-NTH-ROOT-008 The angle between two consecutive sixth roots of any nonzero number is:
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