square root calculator

The Square Root Calculator offers multiple modes to handle different root operations — from basic square roots to higher-order roots. Each mode uses a unique formula to deliver precise results with decimal or radical outputs.

• Square Root (√x): The standard mode — finds the principal square root of any number (e.g., √49 = 7). Ideal for quick math, geometry, or algebra problems.
• Cube Root (³√x): Calculates the cube root for positive or negative numbers (e.g., ³√(-8) = -2). Common in physics and volume-related equations.
• Nth Root (ⁿ√x): Lets you define any integer root (n ≥ 2), such as ⁴√256 = 4 or ⁵√243 = 3. Perfect for advanced calculations and learning exponents.

Bonus Options: Adjust Precision (number of decimal places) and toggle Simplify radicals for integer-only results when you prefer symbolic simplification like √50 = 5√2.

Here’s a quick reference sheet covering the most useful square root formulas — ideal for students, engineers, and math learners.

• Multiplication: √x × √y = √(x·y)
• Square property: (√x)² = x
• Division: √(x/y) = √x ÷ √y
• For decimals: √(0.25) = 0.5
• Perfect squares: √4 = 2, √9 = 3, √16 = 4, √25 = 5

💡 Tip: Use these rules to simplify radicals before converting to decimals — for example, √50 = 5√2 ≈ 7.071.

There are several ways to find the square root of a number — whether by hand, by logic, or with a calculator. Here are the most common manual methods used in math practice:

• 1. Factorization method: Break the number into prime factors, then pair them to find the root.
  Example: √36 = √(6×6) = 6
• 2. Long Division method: A step-by-step approach for large numbers. Divide and average in pairs to get the root’s digits manually.
• 3. Approximation / Calculator: For non-perfect squares, use decimal rounding or a scientific calculator to get accurate results.
• Example: √50 = √(25×2) = 5√2 ≈ 7.071

💡 Tip: Simplify radicals first to keep expressions clean — for example, √72 = √(36×2) = 6√2.

The square root is one of the most widely used mathematical functions — it appears in nearly every technical and scientific field. Here are the most common real-world applications:

• Geometry: Used to calculate side lengths, diagonals, and distances between points — essential in design, construction, and architecture.
• Physics: Appears in speed, energy, and motion formulas — especially in RMS (root mean square) calculations for alternating currents or waveforms.
• Statistics: Used to compute standard deviation and variance, measuring how data points differ from the mean.
• Programming & Computing: Common in algorithms, 3D rendering, graphics, and optimization formulas where distances or magnitudes are required.

💡 Tip: The square root is at the heart of the Pythagorean theorem (a² + b² = c²) — used in both geometry and real-world physics problems.

1. What is a square root?

The square root of a number is the value that, when multiplied by itself, equals the original number. For example, √25 = 5 because 5 × 5 = 25.

2. What is the square root symbol?

The square root is represented by the radical sign √, pronounced “root of.” It may also appear as an exponent: x^½.

3. What is the square root of 1024?

The square root of 1024 is 32 because 32 × 32 = 1024. It’s a perfect square often used in computing and binary powers (2¹⁰).

4. How do you find the square root manually?

You can use factorization, long division, or estimation. For example: √50 = √(25×2) = 5√2 ≈ 7.071.

5. What is the formula of square roots?

The basic relation is x = (√x)². It can also be written as √x = x^½ — the inverse of squaring a number.

6. Is square root same as power ½?

Yes. Mathematically, √x = x^½. Both mean “take the number that multiplied by itself gives x.”

7. What are perfect squares?

Perfect squares are numbers whose square roots are whole numbers — e.g., 1, 4, 9, 16, 25, 36, 49, 64, etc.

8. What is the square root of negative numbers?

Negative numbers don’t have real square roots because no real number squared equals a negative. Their roots are imaginary, written as i√|x| (where i = √−1).

• Use the reference tables above to double-check approximate square root values — helpful for quick mental math.
• If your result shows repeating decimals, round to two decimal places for clean, readable answers.
• Remember: negative numbers don’t have real square roots; their roots are imaginary (i√|x|).
• The absolute value rule: √(x²) = |x| — meaning the result is always positive, even if x is negative.

💡 Tip: Use the calculator’s “Simplify” mode to convert long decimals back into symbolic forms like √50 → 5√2.

Explore more free online math tools to continue learning and solving advanced root and power equations:

• Cube Root Calculator → for ³√x and cubic equations.
• Square Calculator → instantly find x² values and perfect squares.
• Power Calculator → calculate any exponent or nth power.
• Root Mean Square Calculator → for signal, voltage, and physics computations.

🔗 More on square roots and radicals —
Wikipedia.