What is a square root?

For a non-negative real number a , the principal square root √ a is the non-negative number whose square equals a (OpenStax Prealgebra 2e, section 9.1). Every positive number has two square roots: the principal root and its negative. The √ symbol always refers to the principal root.

Can I use negative numbers with this sqrt calculator?

Yes. The calculator accepts negative numbers and returns an imaginary result using the imaginary unit i . √(- a ) = i √ a , where i = √-1 (OpenStax Intermediate Algebra 2e, section 8.8). Enter -9 and you get 3 i .

What does simplified radical form mean?

Simplified radical form expresses a square root by factoring out the largest perfect square. For example, √72 = √(36 × 2) = √36 × √2 = 6√2. The calculator checks whether your input is a perfect square and labels the result accordingly.

Square Root Calculator: Decimal and Simplest Radical Form

Q: Can I control the precision of the decimal result?

Yes. The decimal-places dropdown lets you choose from 0 to 10 decimal places. For most school problems, 2 to 4 decimal places is enough. The decimal you see is an approximation, rounded to the number of places you chose. √2 is not 1.414213562, it is the number that, when squared, gives exactly 2. No finite decimal equals that number. The simplified radical form (√2 itself) is the only exact representation the calculator can provide. The most common mistake is treating the decimal as exact. If you need a measurement to cut a board, use the decimal to the precision your tape measure can read. The calculator gives you both, but confusing them leads to errors that propagate through any calculation that follows.

Square Root Calculator

Calculate the square root of any real number. Perfect squares give an exact answer; other numbers give a decimal approximation and the simplified radical form (√72 = 6√2, √0.5 = √2/2). Negative numbers give an imaginary result (√-16 = 4i).

Input Value

Number:

Display Options

Decimal Places:

Show calculation steps

Show simplified radical form

About Square Roots

A square root of a number is a value that, when multiplied by itself, gives the original number. Every positive number has two square roots: one positive (principal square root) and one negative. By convention, the square root symbol (√) refers to the positive square root.

Key Concepts

Perfect Squares: Numbers whose square roots are integers (e.g., 4, 9, 16, 25).
Irrational Square Roots: Square roots of non-perfect squares result in irrational numbers (e.g., √2, √3).
Principal Square Root: The positive square root of a number, denoted by √.
Simplified Radical Form: Expressing square roots in their simplest form by factoring out perfect squares.

Properties of Square Roots

√(a × b) = √a × √b
√(a/b) = √a / √b (where b ≠ 0)
√a² = |a| (absolute value of a)
(√a)² = a (for a ≥ 0)
√0 = 0
√1 = 1

Applications

Square roots are fundamental in mathematics and have numerous practical applications:

Geometry: Calculating distances and dimensions using the Pythagorean theorem
Physics: Determining velocities, accelerations, and wave frequencies
Statistics: Computing standard deviations and variance
Engineering: Analyzing electrical circuits and structural loads
Finance: Calculating volatility and risk metrics
Computer Graphics: Distance calculations and vector normalization

Every positive number has two square roots: a positive root, called the principal square root, and a negative root. The square root of a negative number is not a real number; it is an imaginary number defined by the imaginary unit i. The square root calculator returns both the decimal approximation and, where possible, the exact simplified radical form. Enter any real number, and you get the principal square root, the simplified radical expression, and a verification step that squares the result to confirm it.

What It Finds: The principal square root of any real number, both as a decimal and in simplest radical form.
Perfect Squares: Give an exact integer result. The calculator confirms the number is a perfect square.
Non-Perfect Squares: Return a decimal approximation and the simplified radical expression (e.g., √72 = 6√2).
Negative Numbers: Return an imaginary result using the imaginary unit <em>i</em> (e.g., √-16 = 4i).
Zero and One: √0 = 0 and √1 = 1, by definition of the principal square root (OpenStax Prealgebra 2e, section 9.1).

How to Use the Square Root Calculator

Type any real number into the input field. Use a decimal point for fractions, a leading minus sign for negatives. The calculator accepts numbers up to roughly 10^300 and as small as 10^-300. Numbers outside that range trigger a warning.

Set the number of decimal places you want in the result: 0 to 10. The display defaults to 2. If the exact square root has fewer decimal places than you requested, trailing zeros are added.

Two optional checkboxes control extra output. Show calculation steps displays the step-by-step reasoning: whether the number is a perfect square, the approximation, and the simplification. Show simplified radical form reveals the exact radical expression even when the decimal is an approximation.

Click Calculate or press Enter. The results section scrolls into view with the square root, the original number for reference, a verification square, the perfect-square check, and the simplified radical form. Click Reset to clear everything and start over.

Reading the Result: Decimal, Radical Form, and Perfect Square Check

The results panel gives you everything in one place.

Square Root Value

This is the principal square root, shown as a decimal rounded to the chosen number of places. If the number is a perfect square, the value is exact and labelled Exact value (Perfect Square). For non-perfect squares, the label says Decimal approximation. For negative inputs, the label adds (imaginary) and the result includes the i suffix.

Simplified Radical Form

When you check the simplified-radical checkbox, the calculator displays the exact expression. For √72 you see 6√2, not a decimal. For √0.5 you see √2/2, because the decimal 0.5 is the fraction 1/2 and the calculator rationalises the fraction before simplifying. This is the simplest radical form: no perfect square factors remain under the radical, no fractions are under the radical, and no radicals appear in the denominator (OpenStax Intermediate Algebra 2e, section 8.2).

Perfect Square Check

The panel answers Is Perfect Square? with Yes or No. A perfect square is an integer whose square root is also an integer, 0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, and so on. If the answer is Yes, the decimal value is exact and the simplified radical form is just that integer.

Verification

The calculator squares the result and shows the product. For exact values this equals the original number exactly. For decimal approximations it gives a number close to the original, confirming the calculation is consistent.

Examples: √50, √72, √0.25, √-9

Four examples show the calculator's full range.

√50

50 is not a perfect square. The decimal approximation to 2 decimal places is 7.07. The simplified radical form is 5√2, because 50 = 25 × 2 and √25 = 5. The verification squares 7.07 to get 50.00 (rounded). The perfect-square check returns No.

√72

72 is not a perfect square. The decimal is 8.49 (2 places). The simplified radical form is 6√2, because 72 = 36 × 2 and √36 = 6. The largest perfect square factor, 36, is pulled out of the radical. This is the core of radical simplification: factor the radicand into a perfect square times a square-free remainder.

√0.25

0.25 is 1/4 as a fraction. Both the numerator (1) and denominator (4) are perfect squares, so the square root is exactly 0.5. The calculator shows Exact value (Perfect Square), and the simplified radical form is 1/2. The verification squares 0.5 to give exactly 0.25.

√-9

A negative radicand has no real square root. The calculator returns 3i, using the imaginary unit i = √-1 (OpenStax Intermediate Algebra 2e, section 8.8). The result label says Decimal approximation (imaginary). The verification squares 3i to get -9, because (3i)² = 9 × i² = 9 × (-1) = -9.

Square Root Examples and Results

Input	Decimal (2 dp)	Simplified Radical Form	Perfect Square?
50	7.07	5√2	No
72	8.49	6√2	No
0.25	0.50	1/2	Yes
-9	3i	3i	No (negative)
16	4.00	4	Yes
0	0.00	0	Yes

Why Most Square Roots Are Irrational

Most integers are not perfect squares. Of the integers from 1 to 100, only ten are perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100. The square roots of the other ninety are irrational numbers, they cannot be expressed as a ratio of two integers. Their decimal expansions never terminate and never repeat.

This is not a limitation of the calculator. It is a property of the number system. √2, for example, is approximately 1.414213562, but no finite decimal captures it exactly. The calculator's decimal is an approximation, not the exact value. The simplified radical form (√2 itself) is the only exact representation. This distinction matters in algebra and geometry: using 1.414 for √2 in a proof loses information that the radical expression preserves.

Irrational square roots appear constantly in practical calculations: the diagonal of a 1×1 square is √2, the side length of a square of area 3 is √3, and the period of a simple pendulum depends on √L. The calculator's radical form gives the exact symbolic answer, which you can then evaluate numerically to any precision you need.