Radical (Root)

A radical is the symbol $\sqrt{\ }$ used to denote a root. The expression $\sqrt[n]{a}$ asks "what number, raised to the $n$-th power, gives $a$?"

• $\sqrt{a} = a^{1/2}$ — square root.
• $\sqrt[3]{a} = a^{1/3}$ — cube root.
• $\sqrt[n]{a} = a^{1/n}$ — n-th root.

Key facts:

• $\sqrt{a^2} = |a|$ — always non-negative for square roots in the reals.
• Even-index roots of negatives are not real (they live in the complex numbers).
• Radicals follow rules like $\sqrt{ab} = \sqrt{a}\sqrt{b}$ and $\sqrt{a/b} = \sqrt{a}/\sqrt{b}$ (for $a, b \geq 0$).

Solving radical equations like $\sqrt{x + 1} = 3$ involves squaring both sides, but you must check for extraneous solutions introduced by squaring (which can flip signs and create false roots).