4.1 nth Roots

Notes

An nth root of $a$ is a number that gives $a$ when raised to the $n$th power. We write $\displaystyle \sqrt[n]{a}$; $n$ is the index.

• $\displaystyle \sqrt[3]{8} = 2$ because $2^3 = 8$. $\displaystyle \sqrt[3]{-27} = -3$ because $(-3)^3 = -27$.
• Odd index: every real number has exactly one real root, positive or negative.
• Even index: positive numbers have two real roots ($\pm$), and $\displaystyle \sqrt[n]{a}$ means the positive one. Negative numbers have no real even roots.
• Example: $x^4 = 81$ has real solutions $x = \pm 3$.