Square Roots & Perfect Squares
Undoing a square, and why √ always hands back the positive answer.
The explanation
Squaring multiplies a number by itself: 6² = 36. A square root runs that backwards: √36 = 6.
Perfect squares are the numbers that come out clean: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100. Knowing these on sight saves enormous time later in algebra.
Numbers in between still have roots, they are just not whole. √20 is a little under 4.5, since 4² = 16 and 5² = 25. If it does not come out clean, you either leave it as √20 or round it.
The radical symbol denotes the principal square root: √a is defined as the unique non-negative b with b² = a, for a ≥ 0. This single-valued definition is what allows √ to be a function.
The distinction that trips people later: the equation x² = 36 has two solutions, x = 6 and x = −6, and you write x = ±6. But the expression √36 has one value, 6. The ± in solving quadratics is something you introduce when taking roots of both sides, not something the radical already contains.
A useful identity is √(x²) = |x|, not x, since the output must be non-negative. Square roots of negatives have no real value, which is exactly the gap that imaginary numbers fill in Algebra 2.
Worked example
Evaluate √81, √(−5)², and solve x² = 49.
- √81 = 9, since 9² = 81.
- √(−5)² = √25 = 5, which equals |−5|.
- x² = 49 has two solutions when you take roots of both sides.
Answer: 9, 5, and x = ±7
Common mistakes
- Writing √36 = ±6. The radical alone is positive; the ± appears when solving an equation.
- Thinking √(a + b) = √a + √b. Try a = 9, b = 16: √25 = 5, not 3 + 4 = 7.