Exercise · ZW-COMPLEX-NTH-ROOT-010 · ★★★ Advanced
The fourth roots of \(-4\) are \(\pm1\pm i\). If \(z\) is one of them, which statement is true?
Choose one option.
Show answer
- B. \(iz\) is also a fourth root of \(-4\)
Explanation
Multiplying by \(i\) rotates by \(\frac\pi2\), the step between consecutive fourth roots, and \(i^4=1\) preserves \(z^4\). Also \(|z|=\sqrt2\), not 2, and \(2z\) would multiply \(z^4\) by 16.
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-007 Find the square roots of i.
- ZW-COMPLEX-NTH-ROOT-008 The angle between two consecutive sixth roots of any nonzero number is: