Exact form first
Shows simplified radical form when the input is an integer with a square factor.
Square Root Calculator
Find the principal square root, exact radical form, decimal value, and clean worked steps from one compact page.
Exact form
Decimal value
Worked steps
Perfect square check
Square Root Calculator
Enter a value, choose the precision, and see the exact form, decimal answer, and verification in the same compact layout.
Popular examples:
✓Summary
Enter your value and press Calculate.
✓Exact
√50 = 5√2
✓Decimal
√50 ≈ 7.07106781
✓Verification
5√2 × 5√2 = 50
✓Rounded result
The decimal result appears here with your selected precision.
✕Perfect Square
No, 50 is not a perfect square.
The calculator always reports the principal square root. Exact radical form is shown whenever the input can be simplified cleanly in real numbers.
- 1.Evaluate the input and identify the radicand.
- 2.Simplify the radical when a perfect-square factor exists.
- 3.Convert the result to decimal form when you need a rounded answer.
Principal Root: The principal root is 5√2.
Negative Root: The second real root is −5√2.
Tip: A perfect square has an integer square root. Use exact form when you want the simplified radical.
What makes this square root calculator useful
This page keeps the calculator close to the top, keeps the answer close to the input, and supports the most common user needs around square roots without making the page feel heavy.
Decimal answer too
Gives a rounded decimal result with your chosen precision for quick checking.
Worked steps
Explains factorisation, simplification, and the final square-root check in plain language.
Readable on one screen
The calculator, result, and steps stay compact so the answer remains close to the input.
How to use the square root calculator
The page is built for quick use first and deeper understanding second, so you can solve the problem immediately and still see the method when you need it.
1) Enter your radicand
Type a whole number, decimal, fraction, or simple expression. Common examples include 50, 2.25, 1/4, 144, or sqrt(50).
2) Choose the precision
Select how many decimal places you want. This is useful when you need a rounded value for homework, measurement, or checking work.
3) Read the answer your way
Use exact form when you want simplified radicals, or switch to decimal form when you need an approximate numeric answer.
A good square root calculator with steps should do more than display a number. It should explain whether the radicand is a perfect square, whether the answer can be simplified into radical form, and how the final number is verified. That is why this page keeps the exact form, decimal answer, and check line together. People searching for an exact square root calculator, a simplify square root calculator, or a square root calculator in decimal form usually want one of those three things first.
Square root formulas and rules
These formulas help with exact simplification, radical manipulation, and understanding what the calculator is doing behind the result.
Basic square root rule
√x = y where y² = x
A square root is the number that gives the original value when multiplied by itself.
Product property
√(ab) = √a × √b
Use this rule to pull perfect-square factors out of a radical and simplify the result.
Quotient property
√(a/b) = √a / √b
This helps with fractions such as √(9/4) = 3/2 when both parts are non-negative.
Principal square root
√x ≥ 0
The radical symbol √ always means the principal, or non-negative, square root.
Perfect square pattern
n = k² ⇒ √n = k
If the radicand is a perfect square, the square root is an exact whole number.
Exponent form
√x = x^(1/2)
Square roots can also be written using exponents, which is useful in algebra and science.
When people search for a square root formula, they are often looking for two different things: the definition of a square root and a practical simplification rule. The definition tells you what the answer means. The product and quotient properties help you simplify radicals. The principal-root rule tells you why √16 = 4 rather than ±4. The plus-minus form appears when solving equations like x² = 16, where the complete solution is x = ±4.
Square root calculator examples
These examples cover common searches such as exact square root, simplified radical form, decimal square roots, and fraction-style square roots.
| Input | Exact form | Decimal value | Why it matters |
|---|---|---|---|
| 50 | 5√2 | ≈ 7.07106781 | A common non-perfect-square example used in algebra. |
| 72 | 6√2 | ≈ 8.48528137 | Shows how perfect-square factors simplify the radical. |
| 2.25 | 1.5 | 1.50000000 | A decimal radicand with an exact rational answer. |
| 1/4 | 1/2 | 0.50000000 | A fraction input can return an exact fraction-style result. |
| 98 | 7√2 | ≈ 9.89949494 | Useful for practice with factorisation and radical simplification. |
| 0.04 | 0.2 | 0.20000000 | Shows how square roots work for small decimals. |
Examples matter because many users do not search with a generic query only. They search for phrases such as square root of 50, square root of 72 in simplest radical form, square root of 2.25, or how to simplify √98. Covering those patterns naturally improves clarity for readers and makes the page more useful across a wider mix of real questions.
Perfect squares chart
Memorising the first perfect squares helps you estimate roots quickly and spot exact answers before you simplify anything.
| Number | Square | Number | Square | Number | Square |
|---|---|---|---|---|---|
| 1 | 1 | 2 | 4 | 3 | 9 |
| 4 | 16 | 5 | 25 | 6 | 36 |
| 7 | 49 | 8 | 64 | 9 | 81 |
| 10 | 100 | 11 | 121 | 12 | 144 |
| 13 | 169 | 14 | 196 | 15 | 225 |
| 16 | 256 | 17 | 289 | 18 | 324 |
| 19 | 361 | 20 | 400 | 21 | 441 |
| 22 | 484 | 23 | 529 | 24 | 576 |
| 25 | 625 | 26 | 676 | 27 | 729 |
| 28 | 784 | 29 | 841 | 30 | 900 |
This perfect squares table supports mental estimation. If a number sits between 64 and 81, then its square root sits between 8 and 9. That quick comparison is especially useful when you want to estimate a value before using the calculator, or when you want to judge whether a decimal answer is reasonable.
Common simplified square roots
These are some of the most common radicals learners check when they need a square root calculator in exact form.
| Original radical | Simplified radical form | Decimal approximation |
|---|---|---|
| √8 | 2√2 | ≈ 2.82842712 |
| √12 | 2√3 | ≈ 3.46410162 |
| √18 | 3√2 | ≈ 4.24264069 |
| √20 | 2√5 | ≈ 4.47213595 |
| √27 | 3√3 | ≈ 5.19615242 |
| √32 | 4√2 | ≈ 5.65685425 |
| √45 | 3√5 | ≈ 6.70820393 |
| √50 | 5√2 | ≈ 7.07106781 |
| √72 | 6√2 | ≈ 8.48528137 |
| √98 | 7√2 | ≈ 9.89949494 |
| √108 | 6√3 | ≈ 10.39230485 |
| √200 | 10√2 | ≈ 14.14213562 |
Square roots of decimals
A square root of decimals calculator is useful for measurements, percentages, scaling, and any problem where the radicand is not a whole number.
| Decimal input | Square root | Comment |
|---|---|---|
| 0.01 | 0.1 | Small decimal square root |
| 0.04 | 0.2 | Useful in percentages and proportions |
| 0.09 | 0.3 | Exact decimal result |
| 0.25 | 0.5 | Same as √(1/4) |
| 0.49 | 0.7 | Common perfect decimal square |
| 2.25 | 1.5 | Useful in geometry and scaling |
| 6.25 | 2.5 | Another exact decimal square |
| 12.25 | 3.5 | Helpful for practice with halves |
Decimals often look harder than they really are. Many decimal radicands are just fraction-style values written in decimal form. For example, 0.25 is the same as 1/4, so the square root is 1/2 or 0.5. Showing exact and decimal output together helps users see that connection immediately.
Where square roots are used
Square roots show up in school maths, professional calculations, geometry, engineering, and data work, so a clear explanation matters just as much as the result.
Geometry
Square roots appear in diagonal, distance, and area-to-side calculations, including right triangles and square dimensions.
Algebra
They are used to solve quadratic equations, simplify radicals, and move between exponent and radical forms.
Science and data
Square roots are part of formulas for variance, standard deviation, signal strength, speed, and measurement scaling.
Everyday estimation
They help with comparing areas, scaling layouts, and checking whether a value sits between two perfect squares.
Study and homework
Worked steps help confirm classroom methods such as factorisation and radical simplification.
Exam revision
The quick reference tables make it easier to memorise perfect squares and common simplified roots.
A strong user-facing page should not stop at a calculator box. It should also explain why the concept matters. This helps the page serve people who are learning, revising, comparing methods, or checking whether the result has a real-world meaning. That broader usefulness is also why the page includes formulas, examples, tables, and FAQ content around the calculator rather than hiding everything below the fold.
Frequently asked questions
These answers cover the most common user questions around square roots, exact form, decimals, negative inputs, and simplification.
What does a square root calculator do?
It finds the principal square root of a number or simple expression, shows a decimal answer, identifies perfect squares, and can simplify many integer radicals into exact form such as √50 = 5√2.
What is the principal square root?
The principal square root is the non-negative root shown by the radical symbol √. For example, both 4 and −4 square to 16, but √16 means 4.
How do you simplify a square root?
Factor the radicand into a perfect square multiplied by another factor. Then take the square root of the perfect square and keep the remaining factor under the radical. Example: √72 = √(36 × 2) = 6√2.
Can this square root calculator handle decimals?
Yes. It accepts decimal inputs such as 2.25, 0.49, or 12.25 and shows the corresponding square root. Exact form is shown when the result is exact or when the radical can be simplified clearly.
Can I enter fractions or simple expressions?
Yes. You can enter values such as 1/4, 81 + 19, or sqrt(50). The calculator first evaluates the expression, then finds the square root of the result when that makes sense in real numbers.
What is a perfect square?
A perfect square is a number made by squaring a whole number. Examples include 1, 4, 9, 16, 25, 36, 49, 64, and 81. These numbers have whole-number square roots.
What happens with negative numbers?
In real numbers, negative values do not have a real square root. If you need √−x, you must switch to complex numbers, where the result is written using i.
Why does the page show both exact form and decimal form?
Exact form is best for algebra and clean symbolic work, while decimal form is useful for measurement, estimation, and checking rounded answers. Showing both helps different users solve the same problem more easily.
How can I estimate a square root mentally?
Find the two closest perfect squares around the number. For example, 49 < 50 < 64, so √50 lies between 7 and 8 and is only a little more than 7.
References
These sources were used to cross-check definitions, principal-root conventions, and common calculator-content patterns.
Math is Fun - Squares and Square Roots in Algebra
View source
CalculatorSoup - Square Roots Calculator
View source
Calculator.io - Square Root Calculator
View source
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