Square Roots: Calculation and Properties
The square root of a number n is the value x such that x² = n. Square roots appear throughout mathematics, physics, and engineering — from the Pythagorean theorem to standard deviation to the quadratic formula.
Perfect Squares Reference
- √1=1, √4=2, √9=3, √16=4, √25=5
- √36=6, √49=7, √64=8, √81=9, √100=10
- √121=11, √144=12, √169=13, √196=14, √225=15
Estimating Non-Perfect Squares
√50: between √49=7 and √64=8
Closer to 7: √50 ≈ 7.07
Verify: 7.07² = 49.98 ✓Simplifying Surds
√72 = √(36×2) = 6√2
√200 = √(100×2) = 10√2
√75 = √(25×3) = 5√3Key Properties
- √(a×b) = √a × √b
- √(a/b) = √a ÷ √b
- √(a²) = |a| (always positive)
- Square root of a negative is imaginary: √(−1) = i
Cube Root and nth Root
∛n = n^(1/3) | ⁴√n = n^(1/4) | ⁿ√x = x^(1/n)Calculate roots instantly: Free Square Root Calculator
Perfect Squares Reference
- √1 = 1, √4 = 2, √9 = 3, √16 = 4, √25 = 5
- √36 = 6, √49 = 7, √64 = 8, √81 = 9, √100 = 10
- √121 = 11, √144 = 12, √169 = 13, √196 = 14, √225 = 15
- √256 = 16, √289 = 17, √324 = 18, √361 = 19, √400 = 20
Applications
The square root appears in geometry (diagonal of a square: d = a√2), the Pythagorean theorem (c = √(a²+b²)), standard deviation in statistics (σ = √variance), physics (RMS speed of gas molecules, pendulum period ∝ √length), and engineering (resonant frequency of an LC circuit ∝ 1/√LC). The quadratic formula uses a square root. Compound interest calculations use nth roots. Understanding square roots is foundational to calculus and higher mathematics.
Frequently Asked Questions
What is the square root of a negative number?
In real numbers, square roots of negative numbers are undefined — no real number multiplied by itself gives a negative result. In complex mathematics, √(-1) is defined as i (the imaginary unit). Complex numbers a + bi where b ≠ 0 have complex square roots. This extension is essential in electrical engineering, quantum mechanics, and signal processing.
How do I estimate square roots mentally?
Find the nearest perfect squares on either side and interpolate. For √50: √49 = 7 and √64 = 8, so √50 is between 7 and 8. Since 50 is much closer to 49, estimate 7.07. The exact value is 7.0711. For rough estimates, use linear interpolation: 7 + (50-49)/(64-49) = 7 + 1/15 ≈ 7.07.
Is √2 rational or irrational?
√2 is irrational — it cannot be expressed as a fraction p/q where p and q are integers. The ancient Greeks proved this around 500 BC. The decimal expansion of √2 = 1.41421356... continues without repeating. In practice, use 1.414 for quick calculations and 1.41421 for greater precision.