Square Roots – Undoing a Square

A square root undoes squaring: √n asks “what number, multiplied by itself, gives n?” Since 6 × 6 = 36, √36 = 6. Square roots are the reverse operation of the square numbers covered earlier in this section – every square number has a clean, whole-number square root, which is exactly why recognising them (the topic of the Perfect Squares page later in this section) is such a useful shortcut.

The radical symbol √ itself first appeared in print in 1525, in the German mathematician Christoff Rudolff's algebra textbook Coss. Historians believe the symbol likely evolved from a stylised lowercase letter “r”, standing for the Latin word radix, meaning “root” – the same root that gives us the modern English word “radical.” The little horizontal bar (called the vinculum) that extends over the number underneath was added later, to make clear exactly which part of an expression the root applied to.

Evaluating Square Roots

Solving Equations with Square Roots

Real-Life Application

• The quadratic formula: solving quadratic equations uses square roots directly.
• The Pythagorean theorem: finding a triangle's side length requires a square root.
• Statistics: standard deviation is calculated as a square root.

Key Takeaways

Practice: Square Roots