Scientific Notation Calculator

Square Root Calculator

Find exact and decimal square roots, simplify radicals to lowest terms, and get complex results for negative radicands.

Exact radical and decimal outputSimplify radicals to lowest termsComplex results for negative radicands

Calculate square roots, simplify radicals, and review each step with copy and PDF output.

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Supports decimals, fractions, powers, parentheses, pi, e, and nested square roots.

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Result Summary

Exact form: 6sqrt(2)
Decimal form: 8.48528137423857
Scientific notation: 8.48528137424 x 10^0

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Square Root Calculator

A square root calculator finds the principal square root of a number or expression, simplifies radical forms, and handles negative radicands by returning imaginary results. Square roots appear in geometry, physics, and statistics, from the Pythagorean theorem to standard deviation formulas. The calculator accepts integers, decimals, fractions, and expressions, then returns the exact simplified radical and the decimal approximation.

What Is a Square Root?

The square root of a number n is the value x such that x² = n. The principal square root is always the non-negative root, written as √n or n^(1/2). Perfect squares such as 16, 25, and 100 have integer square roots: √16 = 4, √25 = 5, √100 = 10. Imperfect squares such as 2, 3, and 7 have irrational square roots that cannot be written as exact fractions. The simplified radical form of √72 is 6√2, because 72 = 36 × 2 and √36 = 6. Simplifying radicals and finding perfect square factors is standard work in algebra, trigonometry, and calculus.

How to Use the Square Root Calculator

Enter a number, decimal, fraction, expression, or a nested root such as sqrt(50) + 2 into the input field. The calculator accepts sqrt() as a function, powers using the ^ symbol, arithmetic operators, and the constants pi and e. Fractions can be entered using the division operator.

Click Solve. The results panel shows the principal square root as an exact simplified radical (when possible), its decimal approximation to full precision, and a step-by-step solution trace. Enable the complex results option to see imaginary outputs in the form a + bi for negative radicands.

Worked Examples

Examples: √72 = 6√2 ≈ 8.485; √144 = 12 (perfect square); √(1/4) = 1/2; √(−9) = 3i (complex result); √50 + √18 = 5√2 + 3√2 = 8√2; and √(pi) ≈ 1.7725.

Common Square Root Problems

Common tasks include simplifying √200, √75, and √48; solving equations of the form x² = 50; finding hypotenuse length using √(a² + b²) from the Pythagorean theorem; and computing root mean square values in physics and statistics.

Square Root Rules and Properties

These rules and tables list the main properties of square roots used in simplification, arithmetic, and equation solving.

Perfect squares and their roots

A perfect square is an integer whose square root is also an integer. Knowing common perfect squares speeds up radical simplification and mental arithmetic.

Perfect square	Square root	Verification	Common use
25	5	5 × 5 = 25	3-4-5 right triangle
144	12	12 × 12 = 144	Geometry problems

Simplifying radicals by factoring

To simplify √n, find the largest perfect square that divides n. Extract its square root in front of the radical and leave the remaining factor inside.

Radical	Factorization	Simplified form	Decimal
√72	√(36 × 2)	6√2	≈ 8.485
√50	√(25 × 2)	5√2	≈ 7.071

Product rule for square roots

The square root of a product equals the product of the individual square roots: √(a × b) = √a × √b. This rule holds when both a and b are non-negative.

Expression	Apply product rule	Simplified	Notes
√(4 × 9)	√4 × √9 = 2 × 3	6	Direct product
√(3 × 27)	√3 × √27 = √81	9	Combine then simplify

Quotient rule for square roots

The square root of a quotient equals the quotient of the individual square roots: √(a/b) = √a / √b. This applies for non-negative numerators and positive denominators.

Expression	Apply quotient rule	Simplified	Notes
√(16/25)	√16 / √25 = 4/5	0.8	Rational result
√(3/4)	√3 / √4 = √3/2	≈ 0.866	Irrational

Square roots of negative numbers

The square root of a negative number is not a real number. It is expressed as an imaginary number using the imaginary unit i, where i = √(−1). The result has the form a + bi.

Radicand	Imaginary form	Simplified	Notes
−9	√(−1) × √9	3i	Pure imaginary
−7	√(−1) × √7	i√7	Irrational imaginary

Adding and subtracting square roots

Square roots can be combined only when they share the same radicand after simplification. First simplify each radical, then add or subtract like terms.

Expression	Simplified first	Combined	Result
√50 + √18	5√2 + 3√2	8√2	Like radicals combined
√8 − √2	2√2 − √2	√2	Simplified difference

Square root of fractions and decimals

For decimal radicands, apply the quotient rule or convert to a fraction first. Decimal inputs that correspond to exact fractions may produce rational square roots.

Radicand	Method	Result	Decimal approx.
0.25	√(1/4) = 1/2	0.5	Exact
0.5	√(1/2) = √2/2	√2/2	≈ 0.7071

Process

Calculation workflow

Enter radicand

Factor and simplify

Apply sign rules

Return radical and decimal

The calculator reads the radicand, factors out the largest perfect square divisor, extracts it from the radical, evaluates the decimal approximation, and applies imaginary number rules for negative inputs. It returns both simplified radical and decimal forms.

Input formats

Supported formats

Input style	Example
Integer radicand	72
Decimal radicand	0.5
Expression	sqrt(50) + 2
Nested root	sqrt(sqrt(16))

Reference

Common scientific constants

Back to calculator

Constant name	Symbol	Notation value	Unit
Speed of light	c	2.99792458 x 10^8	m/s
Gravitational constant	G	6.67430 x 10^-11	N m^2/kg^2
Avogadro constant	NA	6.02214076 x 10^23	mol^-1

Questions

Calculator FAQ

What is a square root?

A square root of a number n is the value that, when multiplied by itself, equals n. The principal square root is the non-negative root. For example, the square root of 25 is 5 because 5 × 5 = 25.

How do I simplify a square root?

Find the largest perfect square that divides your radicand. Take its square root and place it outside the radical. For example, √72 = √(36 × 2) = 6√2.

What is a perfect square?

A perfect square is an integer that is the square of another integer. The first ten perfect squares are 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100.

What is an irrational square root?

A square root is irrational when it cannot be expressed as a fraction of two integers. √2, √3, √5, and √7 are all irrational because their radicands are not perfect squares.

What is the square root of 2?

√2 is an irrational number approximately equal to 1.41421356. It appears in the diagonal of a unit square and in many geometry and trigonometry formulas.

What is the square root of a negative number?

The square root of a negative number is not a real number. It is an imaginary number expressed using the unit i, where i² = −1. For example, √(−16) = 4i.

Can you add two square roots together?

You can add square roots only if they share the same radicand after simplification. For example, √2 + √2 = 2√2, but √2 + √3 cannot be simplified further.

What is the product rule for square roots?

The product rule states that √(a × b) = √a × √b for non-negative a and b. This lets you separate or combine radicands when simplifying expressions.

How is the square root used in the Pythagorean theorem?

In a right triangle with legs a and b, the hypotenuse c = √(a² + b²). The square root extracts the hypotenuse from the sum of the squared legs.

What does it mean to rationalize a denominator?

Rationalizing removes radicals from a fraction's denominator. Multiply numerator and denominator by the radical in the denominator. For example, 1/√3 = √3/3.

How do I calculate √72?

Factor 72 as 36 × 2. √36 = 6 and √2 stays under the radical. The simplified form is 6√2 ≈ 8.485.

What is the square root of 0?

√0 = 0. Zero times zero equals zero, making 0 a perfect square and its own square root.

What is a nested radical?

A nested radical contains a radical inside another radical, such as √(1 + √2). The calculator evaluates nested expressions from the inside out.

How does the square root calculator handle decimals?

Enter the decimal directly as the radicand. The calculator computes the square root numerically and, when possible, returns an exact radical form. For example, √0.25 = 0.5 exactly.

Is every positive number a perfect square?

No. Most positive integers are not perfect squares. Numbers like 2, 3, 5, 6, and 7 are not perfect squares, so their square roots are irrational.