Square Root Calculator
Find the square root of any number — exact form and decimal with step-by-step work.
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How to Calculate Square Roots
The square root of a number x is the value that, when multiplied by itself, equals x. Denoted √x, it is the inverse operation of squaring. The square root of 25 is 5 because 5² = 25.
Perfect Squares
A perfect square is a number whose square root is an integer: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, and so on. For non-perfect squares, the square root is irrational — it continues infinitely without repeating.
Simplifying Square Roots
To simplify √n: find the largest perfect square factor of n, extract it from the radical. For example, √72 = √(36 × 2) = √36 × √2 = 6√2. The simplified form 6√2 is exact, unlike the decimal approximation 8.485...
Square Root Properties
Key rules: √(a×b) = √a × √b, √(a/b) = √a/√b, (√a)² = a, √a² = |a|. Square roots of negative numbers are imaginary (not real numbers).
Applications
Square roots appear in the Pythagorean theorem (c = √(a²+b²)), the quadratic formula, distance formulas, standard deviation, physics (RMS values), and the calculation of side lengths in geometry.
Worked Example: Simplifying √180
Step 1 — Find the prime factorization of 180: 180 = 4 × 45 = 4 × 9 × 5 = 2² × 3² × 5.
Step 2 — Identify perfect square factors: 4 × 9 = 36 is a perfect square factor of 180 (180 = 36 × 5).
Step 3 — Extract the perfect square: √180 = √(36 × 5) = √36 × √5 = 6√5.
Step 4 — Decimal check: 6√5 ≈ 6 × 2.2361 ≈ 13.416. And indeed √180 ≈ 13.416.
Perfect Squares Reference Table
| n | n² | √(n²) |
|---|---|---|
| 1 | 1 | 1 |
| 4 | 16 | 4 |
| 7 | 49 | 7 |
| 9 | 81 | 9 |
| 12 | 144 | 12 |
| 15 | 225 | 15 |
| 20 | 400 | 20 |
Frequently Asked Questions
√2 ≈ 1.41421356... It is irrational, meaning it cannot be expressed as a fraction. In simplified form it stays as √2.
Numbers whose square root is an integer: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144...
Find the largest perfect square factor: 72 = 36 × 2. So √72 = √36 × √2 = 6√2.
Not in real numbers. The square root of a negative number is imaginary (i√|n|). For example, √(-4) = 2i.
√0 = 0. Zero is its own square root.
They are identical. The square root √x is the same as raising x to the power of 1/2. This generalizes: the cube root is x^(1/3), and the nth root is x^(1/n). Using fractional exponents makes algebraic manipulation easier.
Find the nearest perfect squares: √49 = 7 and √64 = 8. Since 50 is close to 49, √50 is slightly more than 7. Linear interpolation: 7 + (50−49)/(64−49) × 1 = 7 + 1/15 ≈ 7.07. The exact value is ≈ 7.0711.