What is the difference between a square root and a square?

A square (x²) multiplies a number by itself: 4² = 16. A square root (√x) is the inverse operation that finds what number was squared: √16 = 4. They are opposite operations that undo each other.

Why are there two square roots (positive and negative)?

Both positive and negative values produce the same square: 4² = 16 and (-4)² = 16. Therefore, 16 has two square roots: +4 and -4, written as ±4. The principal (main) square root is the positive one.

How do I simplify a square root?

Factor the number under the radical into prime factors, then extract perfect squares. For example, √12 = √(4 × 3) = √4 × √3 = 2√3. Our calculator does this automatically and shows the prime factorization.

What is a perfect square?

A perfect square is a number that is the square of an integer: 1, 4, 9, 16, 25, 36, 49, etc. These numbers have exact integer square roots. For example, √16 = 4 exactly (no decimal).

Can I take the square root of a negative number?

No, not in the real number system. Square roots of negative numbers involve imaginary numbers (using 'i' where i² = -1). However, odd-index roots (cube root, fifth root, etc.) can be calculated for negative numbers.

An nth root is a generalization of square roots. The nth root of a number is a value that, when raised to the power of n, gives the original number. For example, the cube root (n=3) of 27 is 3 because 3³ = 27.

How accurate are the decimal approximations?

Our calculator provides decimal approximations to 8 significant figures, which is sufficient for most practical applications. For perfect squares and roots, exact integer values are shown.

What is prime factorization and why is it useful?

Prime factorization breaks a number into its prime number components. For example, 12 = 2² × 3. This is useful for simplifying radicals because perfect squares (like 2² = 4) can be extracted from under the radical sign.

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Square Root Calculator

Calculate square roots, nth roots, and perfect squares with step-by-step solutions. Simplify radicals, view prime factorization, and see exact and approximate values instantly.

Last updated: January 6, 2026

Calculate the square root (√x) of any positive number. Shows both positive and negative roots, simplified form, and checks if it's a perfect square.

Number

√

Result

Positive Root

+4.00000000

Negative Root

-4.00000000

Is Perfect Square/Root?

Yes

Prime Factorization

16 = 2^4

Explanation

The square root of 16 is exactly 4 because 4 × 4 = 16.

Formula Used

√x = x^(1/2)

Step-by-Step Solution

Identify the number under the radical

√16

Find the prime factorization

16 = 2^4

This is a perfect square

√16 = 4

= 4

Calculate the decimal approximation

√16 ≈ 4

= ±4

This calculator provides accurate mathematical calculations for educational and practical purposes.

Results include both exact values (for perfect squares/roots) and decimal approximations. Radical simplifications are shown when applicable.

557 people find this calculator helpful

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, the square root of 16 is 4 because 4 × 4 = 16. Every positive number has two square roots: one positive and one negative (±4 for 16). The square root symbol (√) is called a radical. Square roots are fundamental in mathematics, appearing in geometry, algebra, physics, engineering, and many real-world applications.

Features of This Calculator

Our square root calculator offers comprehensive functionality:

•Square Root (√x): Calculate the square root of any positive number with exact or approximate results
•Nth Root (ⁿ√x): Calculate cube roots, fourth roots, or any custom root index
•Square (x²): Calculate the square of any number instantly
•Simplified Radicals: Automatically simplify radical expressions (e.g., √12 = 2√3)
•Prime Factorization: View the prime factorization to understand radical simplification

Square Root Formulas

Square Root Definition

If √a = b, then b² = a

Example: √16 = 4 because 4² = 16

Nth Root Definition

If ⁿ√a = b, then bⁿ = a

Example: ³√27 = 3 because 3³ = 27

Radical Simplification

√(a × b) = √a × √b

Example: √12 = √(4 × 3) = √4 × √3 = 2√3

Perfect Square

√(n²) = n

Example: √25 = √(5²) = 5

Why Use This Calculator?

1Step-by-step solutions help you understand the calculation process

2Perfect square detection identifies when roots are exact integers

3Automatic radical simplification shows simplified forms

4Prime factorization visualization for learning number theory

5Support for square roots, nth roots, and squaring operations

How to Use the Square Root Calculator

1
Select your operation mode: Square Root (√x), Nth Root (ⁿ√x), or Square (x²)
2
Enter the number you want to calculate with
3
For nth roots, specify the root index (e.g., 3 for cube root, 4 for fourth root)
4
View the result with both exact and approximate values
5
Review the step-by-step solution and prime factorization to understand the calculation

Tips for Working with Square Roots

💡Perfect squares (1, 4, 9, 16, 25, 36, ...) have exact integer square roots
💡To simplify a radical, factor out perfect squares from under the radical sign
💡Every positive number has two square roots: positive and negative (±)
💡The square root of a negative number is not a real number (it's imaginary)
💡Cube roots and odd-index roots can be calculated for negative numbers

Frequently Asked Questions

Loading calculator...

Square Root Calculator

Calculate square roots, nth roots, and perfect squares with step-by-step solutions. Simplify radicals, view prime factorization, and see exact and approximate values instantly.

Last updated: January 6, 2026

Calculate the square root (√x) of any positive number. Shows both positive and negative roots, simplified form, and checks if it's a perfect square.

Number

√

Result

Positive Root

+4.00000000

Negative Root

-4.00000000

Is Perfect Square/Root?

Yes

Prime Factorization

16 = 2^4

Explanation

The square root of 16 is exactly 4 because 4 × 4 = 16.

Formula Used

√x = x^(1/2)

Step-by-Step Solution

Identify the number under the radical

√16

Find the prime factorization

16 = 2^4

This is a perfect square

√16 = 4

= 4

Calculate the decimal approximation

√16 ≈ 4

= ±4

This calculator provides accurate mathematical calculations for educational and practical purposes.

Results include both exact values (for perfect squares/roots) and decimal approximations. Radical simplifications are shown when applicable.

557 people find this calculator helpful

What is a Square Root?

Features of This Calculator

Our square root calculator offers comprehensive functionality:

•Square Root (√x): Calculate the square root of any positive number with exact or approximate results
•Nth Root (ⁿ√x): Calculate cube roots, fourth roots, or any custom root index
•Square (x²): Calculate the square of any number instantly
•Simplified Radicals: Automatically simplify radical expressions (e.g., √12 = 2√3)
•Prime Factorization: View the prime factorization to understand radical simplification

Square Root Formulas

Square Root Definition

If √a = b, then b² = a

Example: √16 = 4 because 4² = 16

Nth Root Definition

If ⁿ√a = b, then bⁿ = a

Example: ³√27 = 3 because 3³ = 27

Radical Simplification

√(a × b) = √a × √b

Example: √12 = √(4 × 3) = √4 × √3 = 2√3

Perfect Square

√(n²) = n

Example: √25 = √(5²) = 5

Why Use This Calculator?

1Step-by-step solutions help you understand the calculation process

2Perfect square detection identifies when roots are exact integers

3Automatic radical simplification shows simplified forms

4Prime factorization visualization for learning number theory

5Support for square roots, nth roots, and squaring operations

How to Use the Square Root Calculator

1
Select your operation mode: Square Root (√x), Nth Root (ⁿ√x), or Square (x²)
2
Enter the number you want to calculate with
3
For nth roots, specify the root index (e.g., 3 for cube root, 4 for fourth root)
4
View the result with both exact and approximate values
5
Review the step-by-step solution and prime factorization to understand the calculation

Tips for Working with Square Roots

💡Perfect squares (1, 4, 9, 16, 25, 36, ...) have exact integer square roots
💡To simplify a radical, factor out perfect squares from under the radical sign
💡Every positive number has two square roots: positive and negative (±)
💡The square root of a negative number is not a real number (it's imaginary)
💡Cube roots and odd-index roots can be calculated for negative numbers

Square Root Calculator

What is a Square Root?

Features of This Calculator

Square Root Formulas

Why Use This Calculator?

How to Use the Square Root Calculator

Tips for Working with Square Roots

Frequently Asked Questions

What is the difference between a square root and a square?

Why are there two square roots (positive and negative)?

How do I simplify a square root?

What is a perfect square?

Can I take the square root of a negative number?

What is an nth root?

How accurate are the decimal approximations?

What is prime factorization and why is it useful?

Square Root Calculator

What is a Square Root?

Features of This Calculator

Square Root Formulas

Why Use This Calculator?

How to Use the Square Root Calculator

Tips for Working with Square Roots

Frequently Asked Questions

What is the difference between a square root and a square?

Why are there two square roots (positive and negative)?

How do I simplify a square root?

What is a perfect square?

Can I take the square root of a negative number?

What is an nth root?

How accurate are the decimal approximations?

What is prime factorization and why is it useful?