√ Square Root Calculator

What does this square root calculator do?

This calculator evaluates the square root of the entered number within the real-number setting. For a positive number, it shows the principal square root and can also explain the positive and negative root pair.

• For positive and decimal inputs, it calculates the square root.
• For positive inputs, it explains the +√x and −√x roots of r² = x.
• For 0, it shows √0 = 0 and the identity 0 = 0².
• For negative inputs, it explains that there is no real square root.
• For non-negative integer inputs, it can show whether the number is a perfect square.

What is a square root?

A square root of a number is a value that gives the original number when squared. For example, 12² = 144, so the principal square root of 144 is 12.

The symbol √x normally means the principal, non-negative square root. That is why √144 is 12, while −12 is the other real solution of the equation r² = 144.

• √144 = 12
• √2 is approximately 1.4142135624
• √2.25 = 1.5
• √0 = 0

Principal square root vs positive and negative roots

The principal square root √x is the non-negative square root when x is not negative. The equation r² = x, however, has two real roots when x is positive: +√x and −√x.

For example, √144 = 12. But the equation r² = 144 has two real solutions: r = 12 and r = −12. Keeping this distinction clear prevents one of the most common square-root mistakes.

What is a perfect square?

A perfect square is a non-negative integer that can be written as the square of an integer. Examples include 0, 1, 4, 9, 16, 25, and 144.

When the input is a non-negative integer, the calculator may show the perfect-square status. For 144, it can show 144 = 12²; for 2, it can state that the number is not a perfect square.

• 144 is a perfect square because 144 = 12².
• 50 is not a perfect square; its square root is approximate.
• 0 is a perfect square because 0 = 0².
• Negative integers are not perfect squares in the ordinary real-number context.

How are decimal square roots handled?

Decimal inputs are supported. For example, √2.25 = 1.5 and √0.25 = 0.5.

Not every decimal square root ends cleanly. In those cases, the displayed value may be approximate. For scientific reporting or measurement work, separate rounding and significant-figure rules may still matter.

What happens with negative numbers?

Negative numbers do not have real square roots. The square of any real number is never negative.

For √-9, the calculator does not show a normal real-number result. It explains that the real square root is undefined. Complex numbers may be mentioned as a mathematical topic, but this calculator does not compute complex roots.

Square root, cube root, and radical simplification

A square root asks for a value whose square is x, while a cube root asks for a value whose cube is x. Radical simplification is a different task, such as rewriting √72 as 6√2.

• Square root: calculates values such as √144 = 12.
• Cube root: calculates values such as ∛−27 = −3; negative inputs can have real cube roots.
• Radical simplification: rewrites expressions such as √72 = 6√2.
• Perfect-square checking: tests whether an integer can be written as n².

How to use the calculator

1. Enter the number whose square root you want to calculate.
2. Use decimal input if needed, such as 2.25 or 0.25.
3. Use the sign control when available if you want to test a negative input.
4. Read the result area for √x, the ± roots when applicable, and the perfect-square status.

Examples

• √144 = 12, and the real roots of r² = 144 are +12 and −12.
• √2 is approximately 1.4142135624.
• √2.25 = 1.5.
• √0 = 0, and 0 = 0².
• √-9 has no real square root; this calculator does not compute complex roots.

Common mistakes

• Treating √144 as ±12. The symbol √144 gives 12; ±12 are the roots of the equation r² = 144.
• Expecting a real square root for a negative number. Negative numbers do not have real square roots.
• Expecting radical simplification. This calculator does not rewrite √72 as 6√2.
• Expecting perfect-square status for every decimal input. Perfect-square checks are meaningful for suitable integer inputs.
• Treating approximate output as unlimited-precision scientific software. Some values may be rounded for display.

Limitations

This calculator works with real-number square roots and basic educational explanations. It does not solve equations, simplify radicals, draw graphs, or compute complex-number roots.

For very large inputs or non-terminating results, displayed values may be rounded or approximate. The calculation does not use official data, APIs, or external datasets.