Square Root Calculator – Compute √ of Any Number
Find the square root of any number instantly with our free online square root calculator. Supports both perfect and imperfect squares with high precision decimal results.
Enter any positive number to find its square root
Understanding Square Roots
The square root of a number is a value that, when multiplied by itself, gives the original number. If 5 × 5 = 25, then √25 = 5. It's the inverse operation of squaring – like how subtraction undoes addition.
Perfect squares like 16, 25, or 144 give clean integer answers. But most numbers have irrational square roots – decimals that go on forever without repeating. This calculator shows up to 10 decimal places of precision and even simplifies radicals when possible.
Square Root Methods
What is a Square Root?
The square root of a number is a value that, when multiplied by itself, gives the original number.
If x² = n, then x = √nPerfect Squares
Numbers whose square roots are whole numbers. These are easy to recognize and calculate.
121, 144, 169, 196, 225, 256, 289, 324, 361, 400
Irrational Square Roots
Most numbers have square roots that are irrational – they can't be written as exact fractions and their decimals never end or repeat.
√2 ≈ 1.4142135623...Simplifying Radicals
Factor out perfect squares from under the radical to simplify.
√72 = √(36 × 2) = 6√2Worked Examples
Example 1: Perfect square
Problem: Find √144
Solution: 144 is a perfect square. 12 × 12 = 144.
Answer: √144 = 12
Example 2: Simplifying a radical
Problem: Simplify √72
Solution: Factor 72 = 36 × 2. Since 36 is a perfect square:
√72 = √(36 × 2) = √36 × √2 = 6√2
Decimal: 6√2 ≈ 6 × 1.414 ≈ 8.485
Example 3: Estimation method
Problem: Estimate √50 without a calculator
Solution: 50 is between 49 and 64. √49 = 7 and √64 = 8.
50 is much closer to 49, so √50 is just above 7.
Actual: √50 ≈ 7.071
Example 4: Large perfect square
Problem: Find √2500
Solution: 2500 = 25 × 100. Both are perfect squares.
√2500 = √25 × √100 = 5 × 10 = 50
Check: 50² = 2500 ✓
Example 5: Complex square root
Problem: Find √(-16)
Solution: Negative numbers don't have real square roots.
In complex numbers: √(-16) = √16 × √(-1) = 4i
Where i is the imaginary unit (i² = -1)
Quick Fact
The square root symbol √ was first used by German mathematician Christoph Rudolff in 1525. It evolved from a stylized letter "r" for "radix" (Latin for "root"). Before this symbol, mathematicians wrote out "radix" or used abbreviations. The horizontal bar over the radicand (the number under the root) was added later by René Descartes in 1637.
How to Calculate Square Roots by Hand
Method 1: Prime Factorization
- Break the number into prime factors
- Group factors into pairs
- Take one from each pair outside the radical
- Multiply remaining factors stay inside
Method 2: Estimation
- Find perfect squares around your number
- Estimate where your number falls between them
- Refine by squaring your guess
- Adjust up or down based on result
Try 7.1: 7.1² = 50.41 (close!)
Try 7.07: 7.07² ≈ 49.98 (very close)
Method 3: Long Division Method
A systematic algorithm similar to long division. Works for any number, gives as many decimal places as needed. Best learned with visual demonstration.
Square Root Properties
Product Rule
√(a × b) = √a × √b√(16 × 9) = √16 × √9 = 4 × 3 = 12
Quotient Rule
√(a / b) = √a / √b√(100 / 4) = √100 / √4 = 10 / 2 = 5
Square Root of a Square
√(a²) = |a|√(5²) = 5, but √((-5)²) = 5 (always positive)
Exponent Form
√a = a^(1/2)Square root equals raising to the power of 1/2
Frequently Asked Questions
What is the square root of a negative number?
In real numbers, you can't take the square root of a negative. But in complex numbers, √(-9) = 3i, where i is the imaginary unit (i² = -1).
Why simplify radicals?
Simplified form makes calculations easier and answers cleaner. 6√2 is more useful than √72 when doing algebra. It's the standard form teachers expect.
Is √2 a rational number?
No. √2 is irrational – it can't be written as a fraction of integers. The ancient Greeks proved this around 500 BCE. The decimal never ends or repeats.
Can a square root have two answers?
Every positive number has two square roots: one positive and one negative. √25 = 5 (principal root), but -5 is also a square root since (-5)² = 25. The radical symbol √ means the positive root only.
How do I type square root on my keyboard?
Use the √ symbol (Alt+251 on Windows, Option+V on Mac), or write sqrt(x), or use exponent notation: x^(1/2). In most math software, sqrt() is the standard function name.
What's the square root of 0?
√0 = 0. Zero is the only number whose square root equals itself. It's also the boundary – positive numbers have real square roots, negative numbers don't (in the real number system).
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