Square Root & Radical Calculator
Calculate square roots, simplify radicals into a√b form, and solve Pythagorean hypotenuses.
e.g. 72 → √72 = 6√2 ≈ 8.485
e.g. 3 for cube root
Result (Decimal)
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Simplified Radical Form
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Is Perfect Square?
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What is the Online Square Root Calculator?
The online Square Root Calculator calculates primary square roots (√x), nth roots, and radical simplifications directly in your browser. When you enter a positive number, the tool outputs both its decimal approximation and its simplest radical form (a√b, such as √50 = 5√2) using prime factor reduction. It also identifies perfect squares, making it useful for algebra, geometry, and engineering checks.
Whether you are checking algebra homework, analyzing structural engineering diagonals, or sizing display screens, this calculator delivers zero-latency answers. Featuring built-in memory registers (MC, MR, M+, M−, MS), you can integrate radical solutions into larger engineering and geometric equations effortlessly.
Step-by-Step Guide: How to Calculate Square Roots
- Input Value: Enter the positive number you want to evaluate (e.g.,
72,144, or50.25). - Click Calculate:
- Principal Square Root / Simplest Radical Form: Expresses radical as
a√b(e.g.,√72 = 6√2). - Decimal Value: Multi-digit precision decimal approximation (e.g.,
√72 ≈ 8.485281). - Perfect Square Verification: Flags whether the input is an integer perfect square.
- Principal Square Root / Simplest Radical Form: Expresses radical as
- Use Pythagorean Mode: Enter legs a and b to compute right-triangle hypotenuse
c = √(a² + b²). - Save to Memory: Use MS and M+ to retain roots for subsequent area and volume formulas.
Fundamental Square Root Formulas & Rules
- Definition: If
x² = a(fora ≥ 0), thenx = √ais the principal square root. - Product Rule:
√(a × b) = √a × √b(e.g.,√72 = √(36 × 2) = 6√2) - Quotient Rule:
√(a / b) = √a / √b - Pythagorean Theorem:
c = √(a² + b²)
Practical Real-World Calculation Examples
- Simplifying Radicals:
√48 = √(16 × 3) = 4√3 ≈ 6.9282. - Room Diagonal (Pythagorean Theorem): Floor measuring 3m by 4m:
c = √(3² + 4²) = √(9 + 16) = √25 = 5 meters. - Finding Square Side Length from Area: A plot with an area of 50 m² has side length:
√50 = 5√2 ≈ 7.071 meters.
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Frequently Asked Questions (FAQ)
What is a square root (√)?
A square root of a number is a value that, when multiplied by itself, gives the original number. For example, because 4 × 4 = 16, the principal square root of 16 is √16 = 4.
How do you simplify a radical expression into a√b form?
Factor the number inside the radical into its prime factors, pull out pairs of identical factors as single numbers outside the square root, and multiply the remaining terms inside (e.g., √72 = √(36 × 2) = 6√2).
What is a perfect square?
A perfect square is an integer generated by multiplying another integer by itself (such as 1, 4, 9, 16, 25, 36, 49, 64, 81, 100). Perfect squares simplify to clean whole numbers.
How do you solve for the hypotenuse using the Pythagorean theorem?
In any right triangle, the hypotenuse c is calculated using c = √(a² + b²), where a and b are the two perpendicular legs (e.g., legs of 3 and 4 give √(9 + 16) = √25 = 5).
Can you calculate the square root of a negative number?
Within real numbers, negative numbers do not have real square roots because squaring any real number produces a positive result. Negative roots require imaginary numbers (i = √−1).
Can this calculator handle square roots of decimals?
Yes, it accurately calculates square roots of decimal numbers such as √0.25 = 0.5 and √2.5 ≈ 1.5811388.
What is the decimal precision of the square root results?
Our computation engine utilizes IEEE-754 double precision to deliver up to 10+ accurate decimal digits.
What are the multiplication and division rules for radicals?
√(a × b) = √a × √b and √(a ÷ b) = √a ÷ √b (for instance, √2 × √3 = √6).
Can I use this calculator to measure diagonals for TV screens or construction?
Yes, simply input width and height into the Pythagorean solver to calculate diagonal corner-to-corner measurements instantly.
Is this square root calculator free on mobile devices?
Yes, it is 100% free with no registration required and works seamlessly across iOS and Android mobile browsers.
How do you find square roots without a calculator?
Manual methods include prime factorization for perfect squares, the long-division square root algorithm, or Newton-Raphson approximation iterations.