Square Root Calculator
Calculate the square root of any number instantly with detailed step-by-step explanations.
What is a Square Root?
The square root of a number x is a value y such that y × y = x. For example, √16 = 4 because 4 × 4 = 16. Square roots are fundamental in geometry (Pythagorean theorem), physics, statistics, and algebra. Our calculator handles both perfect squares and irrational roots with precision.
Square Root Definition
√x = y, where y² = xThe square root of x is the number y that when multiplied by itself equals x.
Product Property
√(a × b) = √a × √bThe square root of a product equals the product of square roots. Example: √12 = √(4×3) = 2√3
Quotient Property
√(a / b) = √a / √bThe square root of a quotient equals the quotient of square roots. Example: √(9/4) = 3/2
Square Root of a Square
√(x²) = |x|The square root of x squared equals the absolute value of x.
Examples
Perfect Square: √49
7 × 7 = 49, so √49 = 7 = 7
Non-Perfect Square: √20
√20 = √(4 × 5) = √4 × √5 = 2√5 ≈ 4.472 = ≈ 4.472
Decimal: √2.25
1.5 × 1.5 = 2.25, so √2.25 = 1.5 = 1.5
Large Number: √144
12 × 12 = 144, so √144 = 12 = 12
Frequently Asked Questions
What is a perfect square?
A perfect square is a number that has a whole number as its square root. Examples: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144.
Can you take the square root of a negative number?
Not with real numbers. The square root of a negative number involves imaginary numbers (√-1 = i). For real number calculations, square roots of negative numbers are undefined.
How do I estimate square roots mentally?
Find the two closest perfect squares. For √50: since √49 = 7 and √64 = 8, √50 is slightly more than 7 (actually ≈ 7.07). The answer is always between the two perfect square roots.
What is the principal square root?
The principal (or positive) square root is the non-negative root. While both 4² and (-4)² give 16, √16 = 4 by convention. We always take the positive value.
What is a radical vs a square root?
A radical is the √ symbol. Square root is a specific case where the index is 2 (√x). Other radicals include cube root (∛x, index 3) and fourth root (∜x, index 4).