Use this Square Root Calculator to instantly convert between a number and its principal square root. Enter a number x to find √x, or enter a square root value y to find the original number x using y² = x.

Important Note: This Square Root Calculator works with real-number principal square roots. For a nonnegative number x, it returns the principal nonnegative square root √x. You can also reverse the calculation by entering a square root value y to find the original number using x = y².

This visible calculator does not simplify radicals symbolically and does not calculate complex square roots for negative radicands. For negative inputs, complex-number math is required.

Reviewed by: AjaxCalculators Editorial Team
Last updated: April 28, 2026
Method source: Standard principal square root definition, y = √x, and reverse relationship x = y²
Editorial standards: AjaxCalculators Editorial Policy

What This Square Root Calculator Calculates

This calculator converts between a number and its principal square root. It works in both directions: from number to square root, and from square root back to the original number.

The calculator can calculate:

• Square root of a number
• Principal square root √x
• Original number from a square root value
• Reverse square root calculation using y² = x
• Step-by-step derivation

The visible tool has two input fields: Number (x) and Square root (y). Enter either field to see the matching value and derivation.

What a Square Root Means

A square root of a number is a value that, when multiplied by itself, gives the original number.

For example:

√25 = 5

This is because:

5 × 5 = 25

Using symbols, if:

y = √x

Then:

y² = x

So, the square root and the square are inverse operations.

What Principal Square Root Means

The principal square root is the nonnegative square root of a nonnegative number. This is the value normally meant when the square root symbol √ is used.

For example, both 5 and −5 square to 25:

5² = 25

(−5)² = 25

But the principal square root is:

√25 = 5

It is not written as −5 unless the negative sign is shown separately, such as −√25.

How the Square Root Calculator Works

1) Number to Square Root

If you enter a number x, the calculator finds the principal square root.

y = √x

For example:

√64 = 8

So, if x = 64, then y = 8.

2) Square Root to Number

If you enter a square root value y, the calculator reverses the relationship by squaring y.

x = y²

For example:

9² = 81

So, if y = 9, then x = 81.

3) Exponent Notation

A square root can also be written using exponent notation.

√x = x^1/2

For example:

√36 = 36^1/2 = 6

This is why square roots are also connected to fractional exponents.

Square Root Formula Summary

What You Want to Find	Formula	Meaning
Principal square root	y = √x	Finds the nonnegative square root of x.
Original number from square root	x = y2	Squares the square root value to reverse the calculation.
Exponent form	√x = x1/2	Writes square root as a fractional exponent.
Real-number condition	x ≥ 0	The radicand must be nonnegative for a real square root.
Principal root condition	√x ≥ 0	The radical symbol returns the nonnegative square root.

Worked Example: Find the Square Root

Suppose you want to find the square root of 144.

Step 1: Start with the square root expression
y = √144

Step 2: Find the number that squares to 144
12 × 12 = 144

Step 3: Write the principal square root
√144 = 12

So, the principal square root of 144 is 12.

Worked Example: Reverse Square Root Calculation

Suppose the square root value is 7.5, and you want to find the original number.

Step 1: Use the reverse formula
x = y²

Step 2: Substitute y = 7.5
x = 7.5²

Step 3: Calculate
x = 7.5 × 7.5
x = 56.25

So, a square root value of 7.5 corresponds to the original number 56.25.

Worked Example: Decimal Square Root

Suppose you want to find the square root of 2.25.

Step 1: Start with the expression
y = √2.25

Step 2: Find the nonnegative value that squares to 2.25
1.5 × 1.5 = 2.25

Step 3: Write the answer
√2.25 = 1.5

So, the principal square root of 2.25 is 1.5.

Worked Example: Non-Perfect Square

Suppose you want to calculate √10.

Step 1: Check nearby perfect squares
3² = 9
4² = 16

Step 2: Understand the result range
Since 10 is between 9 and 16, √10 is between 3 and 4.

Step 3: Use decimal approximation
√10 ≈ 3.1623

Step 4: Check by squaring
3.1623² ≈ 10

So, √10 is approximately 3.1623.

How to Use This Square Root Calculator

1. Enter a value in the Number (x) field to find its square root.
2. Or enter a value in the Square root (y) field to reverse-calculate the original number.
3. Use only one field at a time for the clearest result.
4. Review the converted value in the other field.
5. Check the step-by-step derivation to understand the calculation.
6. Click Reload calculator to clear the fields and start again.

How to Interpret the Result

Result	What It Means	Example
Number (x)	The original number under the square root symbol.	If x = 49, the expression is √49.
Square root (y)	The principal nonnegative square root of x.	If x = 49, then y = 7.
Reverse result	The original number found by squaring y.	If y = 7, then x = 72 = 49.
Step-by-step derivation	The formula path used by the calculator.	It shows whether the tool used y = √x or x = y2.

Common Square Roots Table

Number	Principal Square Root	Check
0	0	0 × 0 = 0
1	1	1 × 1 = 1
4	2	2 × 2 = 4
9	3	3 × 3 = 9
16	4	4 × 4 = 16
25	5	5 × 5 = 25
36	6	6 × 6 = 36
49	7	7 × 7 = 49
64	8	8 × 8 = 64
81	9	9 × 9 = 81
100	10	10 × 10 = 100

Perfect Squares

A perfect square is a number with a whole-number square root. For example, 64 is a perfect square because:

8² = 64

Other common perfect squares include 1, 4, 9, 16, 25, 36, 49, 81, and 100.

Numbers that are not perfect squares often have square roots that are decimals. For example, √2 is approximately 1.4142, and √3 is approximately 1.7321.

Square Root vs Square

Square root and square are inverse operations. Squaring multiplies a number by itself, while taking the square root finds the principal value that squares back to the original nonnegative number.

Operation	Meaning	Example
Square	Multiply a number by itself.	62 = 36
Square root	Find the principal value that squares to the number.	√36 = 6

If you square a principal square root, you return to the original nonnegative number:

(√x)² = x

For x ≥ 0, taking the square root of x² gives the absolute value of x:

√(x²) = |x|

Perfect Squares vs Non-Perfect Squares

A perfect square has a whole-number square root. A non-perfect square usually has a decimal or irrational square root.

Number Type	Example	Square Root Result
Perfect square	64	√64 = 8
Perfect square	100	√100 = 10
Non-perfect square	2	√2 ≈ 1.4142
Non-perfect square	3	√3 ≈ 1.7321
Non-perfect square	10	√10 ≈ 3.1623

Can You Take the Square Root of a Negative Number?

In the real number system, a negative number does not have a real square root. This is because squaring a positive number gives a positive result, and squaring a negative number also gives a positive result.

For example:

4² = 16

(−4)² = 16

No real number squared gives −16.

Negative square roots can be handled using complex numbers, where the imaginary unit i is defined by i² = −1. However, this visible calculator is best positioned as a real-number square root calculator unless complex-number output is added.

Real vs Complex Square Roots

In the real number system, square roots of negative numbers are not real. Complex numbers are needed to represent square roots of negative values.

Input Type	Real Square Root?	Example
Positive number	Yes	√25 = 5
Zero	Yes	√0 = 0
Negative number	No real square root	√−25 requires complex-number math.

This visible calculator is best positioned as a real-number square root calculator unless complex-number output is added.

Principal Root vs Both Square Roots

For a positive number, there are two values that square to it: one positive and one negative. However, the radical symbol √x normally gives the principal, nonnegative square root.

Expression	Meaning	Example Result
√25	Principal square root only	5
−√25	Negative of the principal square root	−5
±√25	Both square roots	±5

Do not write √25 = ±5. The correct principal square root is √25 = 5. Use ±√25 only when a problem asks for both roots.

When to Use a Square Root Calculator

A square root calculator can be useful when you need to find a root quickly, check a math problem, estimate a side length, or reverse a square.

Common uses include:

• solving basic algebra problems
• checking perfect squares
• finding side length from square area
• using the Pythagorean theorem
• checking geometry calculations
• working with statistics formulas
• calculating standard deviation steps
• checking engineering or physics equations
• converting between x and √x

Square Root in Geometry

Square roots are common in geometry because they help recover a length from an area or from a squared relationship.

Geometry Situation	Formula	Example
Find side length from square area	side = √area	If area = 49, side = √49 = 7.
Find hypotenuse of a right triangle	c = √(a2 + b2)	If a = 3 and b = 4, c = √25 = 5.
Find distance from coordinate differences	d = √((x2 − x1)2 + (y2 − y1)2)	Uses the distance formula in coordinate geometry.

Square Root in the Pythagorean Theorem

The square root is also used in the Pythagorean theorem. For a right triangle:

a² + b² = c²

To find the hypotenuse c:

c = √(a² + b²)

For example, if a = 3 and b = 4:

c = √(3² + 4²)

c = √(9 + 16)

c = √25 = 5

Important Assumptions and Limitations

Assumption or Limitation	What It Means
Principal square root only	The calculator returns the nonnegative principal square root for nonnegative x.
Real-number calculator	For real-number results, the number under the square root should be 0 or greater.
No complex-number output	The visible tool does not calculate complex square roots for negative numbers.
No radical simplification	It does not simplify expressions such as √72 into 6√2.
No equation solving	It does not solve general radical equations or algebraic systems.
No cube roots or nth roots	It is focused on square roots only.
Rounded decimal results	Some non-perfect square results may be rounded for display.
Not a symbolic math tool	Use a radical simplifier or algebra system for exact symbolic results.

Common Mistakes to Avoid

Mistake	Why It Causes Problems
Confusing √x with x2	Square root and square are inverse operations, not the same operation.
Writing √25 = ±5	The radical symbol gives the principal square root, so √25 = 5.
Using ±√x when only the principal root is needed	Use ± only when a problem asks for both square roots.
Entering a negative number for a real square root	Negative radicands require complex-number math.
Assuming every square root is a whole number	Many square roots are decimal or irrational values.
Rounding too early	Early rounding can affect later geometry, algebra, statistics, or physics calculations.
Forgetting that √0 = 0	Zero is a valid square root input and result.
Confusing square roots with cube roots	Square roots use power 1/2, while cube roots use power 1/3.

Practical Uses

This Square Root Calculator can be useful for:

• checking homework answers
• finding the principal square root of a number
• reverse-calculating x from a known square root
• learning the relationship between x and √x
• checking perfect squares
• working with geometry formulas
• using the Pythagorean theorem
• checking statistics and standard deviation calculations
• understanding exponent notation x^1/2

When You May Need a Different Calculator

This calculator is best for simple real-number principal square root calculations and reverse square root calculations. You may need another tool if your problem requires symbolic algebra or complex-number output.

Need	Better Tool or Method
Simplify radicals	Use a radical simplifier.
Solve square root equations	Use an algebra equation solver.
Calculate complex square roots	Use a complex-number calculator.
Find cube roots	Use a cube root calculator.
Find nth roots	Use an nth root calculator.
Work with matrices or advanced algebra	Use a computer algebra system or advanced scientific calculator.

References

1. Wolfram MathWorld — Square Root
2. Wolfram MathWorld — Principal Square Root
3. Khan Academy — Introduction to Square Roots
4. Khan Academy — Understanding Square Roots
5. OpenStax — Radicals and Rational Exponents

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Square Root Calculator Disclaimer

This Square Root Calculator provides principal square root calculations for real-number use. For a nonnegative number x, the principal square root √x is the nonnegative value that squares to x. The calculator can also reverse the relationship by using x = y².

This calculator does not calculate complex square roots for negative inputs, simplify radicals symbolically, solve radical equations, find cube roots, calculate nth roots, or perform advanced algebra. For exact radical simplification, complex-number square roots, or symbolic equation solving, use a tool designed for algebra or complex-number math.