Nth root

Summary

• For \(n∈N\) , t he n th root of a real number \(a\) is a solution to the equation

\[x^n=a\]

• If \(n\) is even then
• For \(a≥0\) , the equation \(x^n=a\) has a unique real non-negative solution
• For \(a<0\) , the equation \(x^n=a\) has no real solution
• For \(a≥0\) , the unique real non-negative solution to the equation is called the principal n th root of \(a\) .
• The principal n th root of \(a\) is denoted by \(\sqrt[n]{a}\) or \(a^{1/n}\)
• If \(n\) is odd then
• The equation \(x^n=a\) has a unique real solution for any \(a\) (which may be negative)
• The n th root of \(a\) is denoted by \(\sqrt[n]{a}\) or \(a^{1/n}\) (which may be negative).