Square Number vs Square Root: 12, 144 and √12

The operations run in opposite directions

Squaring 12 means multiplying it by itself: 12² = 144. Taking the principal square root of 144 asks for the nonnegative value whose square is 144, so √144 = 12. Taking the square root of 12 asks a different question.

Twelve is not a perfect square because it lies between 9 = 3² and 16 = 4². Its square root therefore lies between 3 and 4. Writing √12 = 6 would confuse halving with taking a root.

Simplify before rounding

Since 12 = 4 × 3, √12 = √4 × √3 = 2√3. That expression is exact. Its decimal approximation is about 3.464102 when rounded to six decimal places. An equals sign belongs with the exact expression; an approximation sign belongs with the rounded decimal.

Prime factorization explains the method. Each pair of equal prime factors under a square root contributes one copy outside it. For 72 = 2³ × 3², take out one 2 and one 3, leaving 6√2.

A root symbol and an equation are different

The symbol √144 denotes the principal, nonnegative root 12. The equation x² = 144 has two real solutions, 12 and −12. Number JD’s reference table reports the principal square root.

Use the nearest square numbers as a quick reasonableness check. A claimed root of 72 must lie between 8 and 9 because 64 < 72 < 81. This catches many decimal-entry mistakes before you need a detailed calculation.

Check the mathematical definitions

For fractions, percentages and worked arithmetic, see OpenStax Prealgebra. For number definitions, see OpenStax on prime and composite numbers and Wolfram MathWorld on digital roots. Worked examples on this page are calculated for this guide.