Question

66 give all possible roots to the radical \\(\sqrt3{-8}\\)

Explanation:

Step1: Recall odd - root property

For an odd - degree root \(\sqrt[n]{x}\) where \(n\) is odd, the equation \(y = \sqrt[n]{x}\) is equivalent to \(y^{n}=x\). Also, for a real - valued odd root of a negative number, we know that if \(n\) is odd, \(\sqrt[n]{-a}=-\sqrt[n]{a}\) for \(a\geq0\). And for the cube root, the general formula for the roots of the equation \(y^{3}=x\) (in the complex plane) is given by \(y_{k}=\sqrt[3]{|x|}e^{i(\frac{\arg(x)+2k\pi}{3})}\), where \(k = 0,1,2\) and \(\arg(x)\) is the argument of the complex number \(x\). But first, let's consider the real - valued root and then the complex roots.

First, for the real - valued root: We know that \((- 2)^{3}=-8\), so \(\sqrt[3]{-8}=-2\) in the real numbers.

Now, for the complex roots: Let \(y^{3}=-8 = 8e^{i\pi}\) (in polar form, since \(-8 = 8(\cos\pi+i\sin\pi)\)). Using De Moivre's theorem, the \(n\) - th roots of a complex number \(r(\cos\theta + i\sin\theta)\) are given by \(r^{\frac{1}{n}}(\cos(\frac{\theta + 2k\pi}{n})+i\sin(\frac{\theta + 2k\pi}{n}))\) for \(k = 0,1,\cdots,n - 1\). Here, \(n = 3\), \(r = 8\), \(\theta=\pi\).

So, for \(k = 0\):
\(y_{0}=8^{\frac{1}{3}}(\cos(\frac{\pi+2\times0\times\pi}{3})+i\sin(\frac{\pi + 2\times0\times\pi}{3}))=2(\cos\frac{\pi}{3}+i\sin\frac{\pi}{3})=2(\frac{1}{2}+i\frac{\sqrt{3}}{2}) = 1 + i\sqrt{3}\)

For \(k = 1\):
\(y_{1}=8^{\frac{1}{3}}(\cos(\frac{\pi+2\times1\times\pi}{3})+i\sin(\frac{\pi + 2\times1\times\pi}{3}))=2(\cos\pi+i\sin\pi)=2(- 1 + 0i)=-2\) (this is the real root we found earlier)

For \(k = 2\):
\(y_{2}=8^{\frac{1}{3}}(\cos(\frac{\pi+2\times2\times\pi}{3})+i\sin(\frac{\pi + 2\times2\times\pi}{3}))=2(\cos\frac{5\pi}{3}+i\sin\frac{5\pi}{3})=2(\frac{1}{2}-i\frac{\sqrt{3}}{2})=1 - i\sqrt{3}\)

Step2: List all possible roots

The three cube roots of \(-8\) are:

• When \(k = 0\): \(1 + i\sqrt{3}\)
• When \(k = 1\): \(-2\)
• When \(k = 2\): \(1 - i\sqrt{3}\)

Answer:

The possible roots of \(\sqrt[3]{-8}\) are \(-2\), \(1 + i\sqrt{3}\), and \(1 - i\sqrt{3}\)