Exercise · ZW-COMPLEX-NTH-ROOT-005 · ★★☆ Intermediate
The cube root of \(-8\) with \(k=0\) in the formula is:
Choose one option.
Show answer
- B. \(1+i\sqrt3\)
Explanation
\(r=8\), \(\theta=\pi\): \(2\left(\cos\frac\pi3+i\sin\frac\pi3\right)=1+i\sqrt3\). It satisfies \((1+i\sqrt3)^3=-8\). The real root \(-2\) is the one with \(k=1\).
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-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:
- ZW-COMPLEX-NTH-ROOT-009 Which of these is a cube root of -27i?