Calculate the square root of any number with detailed explanations. Perfect for students, engineers, and quick calculations.
The square root of a number is a value that, when multiplied by itself, equals the original number. It's the inverse operation of squaring a number.
Mathematical notation: √x or x1/2
Example: √144 = 12 because 12 × 12 = 144
Perfect squares are numbers whose square roots are whole numbers (integers). Examples: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, etc.
Non-perfect squares have irrational square roots—decimals that never end or repeat. Examples: √2 = 1.41421356..., √3 = 1.73205080..., √5 = 2.23606797...
Most calculators have a √ button. Enter number, press √ button. Digital calculators and computer software use iterative algorithms (like Newton's method) to approximate square roots to many decimal places instantly.
Break number into prime factors, pair them up, multiply one from each pair.
Example: √144
Example: √180
For mental math or when calculator unavailable:
Example: √50
Before calculators, people used long division-like algorithm to find square roots by hand. Still taught in some schools for understanding, but impractical today. Similar to long division but involves pairing digits and finding largest square at each step. Tedious but works for any number to any precision.
Iterative approximation: Start with guess, refine until accurate.
Formula: Next guess = (guess + number/guess) / 2
Example: √20
30² = 900, 40² = 1,600, 50² = 2,500, 60² = 3,600, 70² = 4,900, 80² = 6,400, 90² = 8,100, 100² = 10,000
Memorizing these helps with mental estimation and recognizing perfect squares in problems.
√(a × b) = √a × √b
Example: √72 = √(36 × 2) = √36 × √2 = 6√2
Use this to simplify square roots by factoring out perfect squares.
√(a / b) = √a / √b
Example: √(25/4) = √25 / √4 = 5 / 2 = 2.5
√(x²) = |x| (absolute value of x)
Example: √(5²) = √25 = 5, and √((-5)²) = √25 = 5
The square root of a squared number always gives the positive (absolute) value.
Every positive number has two square roots: positive and negative. 3² = 9 AND (-3)² = 9. The √ symbol means principal (positive) square root only.
√9 = 3 (not ±3). If you need both roots, write ±√9 = ±3.
Square root of negative numbers involves imaginary numbers (i = √-1). √-16 = 4i. This is complex number territory, not covered by standard square root calculator. In real number system, square roots of negative numbers are undefined.
Square area to side length: Floor area = 324 sq ft. Side length = √324 = 18 feet.
Diagonal of square: Diagonal = side × √2. Square with 10" sides has diagonal = 10√2 ≈ 14.14 inches.
Pythagorean Theorem: Right triangle with legs 3 and 4. Hypotenuse = √(3² + 4²) = √(9 + 16) = √25 = 5. The famous 3-4-5 right triangle.
Distance between two points (x₁, y₁) and (x₂, y₂): d = √((x₂ - x₁)² + (y₂ - y₁)²)
Example: Distance from (1, 2) to (4, 6): d = √((4-1)² + (6-2)²) = √(9 + 16) = √25 = 5 units
Time to fall from height: t = √(2h/g), where h = height, g = gravity (9.8 m/s²)
Example: Object dropped from 80 meters. Time = √(2 × 80 / 9.8) = √16.33 ≈ 4.04 seconds
Finding annual growth rate from total growth over 2 years: Annual rate = √(Final/Initial) - 1
Example: Investment grows from $10,000 to $12,100 in 2 years. Annual rate = √(12,100/10,000) - 1 = √1.21 - 1 = 1.1 - 1 = 0.10 = 10% per year
Standard deviation is square root of variance. Variance = average squared deviation. Standard deviation brings units back to original scale.
Example: Variance = 64. Standard deviation = √64 = 8.
Express square root with smallest possible number under the radical by factoring out perfect squares.
Example: Simplify √200
Example: Simplify √288
Can only add/subtract square roots with same radicand (number under √).
Valid: 3√5 + 2√5 = 5√5 (like terms, same as 3x + 2x = 5x)
Invalid: √2 + √3 cannot be simplified (different radicands)
Example: √50 + √18
Example: √48 - √12 + √75
√a × √b = √(a × b)
Examples:
√a / √b = √(a / b)
Examples:
Eliminate square roots from denominators by multiplying by √/√.
Example: 5/√3
Example: 12/√8
❌ Wrong: √4 + √9 = √13
✓ Correct: √4 + √9 = 2 + 3 = 5
Cannot add numbers inside different square roots. Simplify each first, then add.
❌ Wrong: (√5)² = 5²
✓ Correct: (√5)² = 5
Squaring and square root are inverse operations—they cancel out.
❌ Wrong: √(x + y) = √x + √y
✓ Correct: √(x + y) cannot be simplified this way
Product/quotient properties work, but NOT sum/difference. √(4 + 9) = √13 ≠ 2 + 3.
❌ Wrong: √9 = ±3
✓ Correct: √9 = 3 (principal square root is positive only)
√ symbol means positive root. If equation is x² = 9, then x = ±3. But √9 = 3 only.
❌ Wrong: √24 = 2√6 (missed larger perfect square factor)
✓ Correct: √24 = √(4 × 6) = 2√6, or better: √(4 × 4 × 1.5)... wait, √24 = 2√6 is correct!
Actually the first was right. Better example: √32 = 4√2 not 2√8 (fully simplify).
Not in the real number system. √-1 is defined as "i" (imaginary unit) in complex numbers. √-9 = 3i. For real-world calculations and basic algebra, square roots of negatives are undefined. Scientific calculators show "Error" for √(-x).
They're inverse operations. Squaring (²) multiplies number by itself. Square root (√) finds what number multiplied by itself gives the result. 5² = 25. √25 = 5. They undo each other: √(x²) = x and (√x)² = x.
Proof by contradiction shows √2 cannot be expressed as fraction. It's decimal expansion never ends or repeats: 1.41421356237309504880168872420969... Many square roots are irrational (any non-perfect square). This was one of ancient Greeks' most important mathematical discoveries.
Find nearest perfect squares above and below. For √30: 25 < 30 < 36, so 5 < √30 < 6. Since 30 is closer to 25, estimate ≈5.5. (Actual: 5.477). Practice with perfect squares 1-100 to get faster.
√0 = 0. Zero is special case—only number equal to its own square root. 0 × 0 = 0, so √0 = 0.
Depends on context. Engineering: 3-4 decimals usually sufficient. Scientific calculations: 6-8. Financial: 2. Most real-world situations don't need beyond 4 decimal places. Additional precision is just noise.