Calculate square roots, cube roots, and any nth root. Simplify radicals, find exact and decimal answers, work with negative numbers, and see step-by-step math explaining how roots work.
Choose what you want to calculate, enter your numbers, and see the exact math behind the answer.
Whole numbers, decimals, and simple fractions (like 9/16) are all accepted. Negative numbers are allowed too, they simply have no real square root.
A square root is a number that produces the original number when multiplied by itself. For example: 5×5=25. Therefore: √25=5. The symbol √ is called the radical symbol. The number written inside it is called the radicand. In √25, 25 is the radicand. Square roots undo the operation of squaring a nonnegative number. For example: 12²=144. Therefore: √144=12. This inverse relationship is one of the most important ideas to understand about roots. A square root is not the same thing as dividing a number by 2. For example: √16 is 4, not 8. That is because 4×4=16.
The radical symbol √x represents the principal square root of x. For a nonnegative real number, the principal square root is the nonnegative value whose square equals the original number. For example: √36=6. Why not -6? Because the symbol √36 by itself refers to the principal square root. However: (-6)² also equals 36. This becomes important when solving an equation. If the equation is x²=36, then there are two real solutions: x=6 and x=-6. So: √36=6. But: x²=36 has solutions x=±6. These statements are not contradictory. They answer different mathematical questions.
The square root of 2 is √2. It cannot be simplified to a whole number or ordinary fraction. Its decimal begins 1.41421356... and continues without terminating or repeating. That makes √2 an irrational number. The exact answer is √2. A decimal such as 1.4142 is only an approximation. You can check the approximation: 1.4142² is close to 2. But the exact symbolic value √2 preserves the full mathematical quantity. This is why a useful square-root calculator should show both exact form and decimal approximation.
The square root of 3 is √3. Its decimal value is approximately 1.73205081. Because 3 has no perfect-square factor greater than 1, the radical cannot be simplified further over the integers.
The square root of 4 is 2 because 2²=4. Four is a perfect square.
The exact square root of 5 is √5. Its decimal approximation is about 2.23606798. Like √2 and √3, √5 is irrational. These specific roots are common search questions, but the method is always the same: find the nonnegative value whose square produces the original number.
A perfect square is a number that can be written as the square of an integer. Examples: 1=1², 4=2², 9=3², 16=4², 25=5², 36=6², 49=7², 64=8², 81=9², 100=10². Perfect squares have whole-number square roots. For example: √81=9. Numbers between perfect squares usually do not have whole-number square roots. For instance: 16<20<25. Therefore: 4<√20<5. This kind of estimation can help you quickly tell whether a calculated decimal answer makes sense.
A radical can sometimes be simplified by finding a perfect-square factor. Consider √72. Find the largest perfect square that divides 72. That number is 36. Write 72=36×2. Then √72=√(36×2). Using the product property: √72=√36×√2. Since √36=6, we get √72=6√2. This is the simplified exact answer. Its decimal approximation is about 8.4853. The exact and decimal answers serve different purposes. The radical 6√2 is mathematically exact. The decimal is convenient for measurement and estimation.
You can use any perfect-square factor, but choosing the largest one often simplifies the radical in fewer steps. Example: √200. One factorization is 200=4×50. Then √200=2√50. But 50 can still be simplified. A more efficient choice is 200=100×2. Then √200=√100×√2=10√2. Either valid path eventually reaches the same simplified answer.
A cube root asks: "What number multiplied by itself three times gives this number?" The symbol is ∛. Example: ∛27. Because 3×3×3=27, ∛27=3. Cube roots undo cubing. Example: 5³=125. Therefore: ∛125=5.
Yes. Odd roots of negative real numbers can be real. Example: ∛(-8). Because (-2)×(-2)×(-2)=-8, ∛(-8)=-2. The situation is different for square roots. There is no real number whose square is -16. Therefore: √(-16) has no real-number result. In complex numbers: √(-16) has principal value 4i. This is why the root index matters. Odd roots can preserve a negative sign. Even roots of negative real numbers require complex numbers.
An nth root generalizes square roots and cube roots. The expression n√x asks: "What number raised to the nth power equals x?" Examples: 2√16=4. 3√27=3. 4√16=2. 5√32=2. The small number n is called the root index. When no index is shown on the radical symbol √x, the index is understood to be 2.
Roots can be written using fractional exponents. Formula: n√x=x(1/n). Therefore: √x=x(1/2). Cube root: ∛x=x(1/3). Fourth root: 4√x=x(1/4). Example: √81=81(1/2)=9. Example: ∛27=27(1/3)=3. The Exponent Calculator can help when your main problem involves powers, negative exponents, fractional exponents, or solving for a missing exponent.
Square roots can be applied to fractions. Example: √(9/16). Take the square root of the numerator and denominator: √9/√16. That gives 3/4. Verification: (3/4)²=9/16. Not every fraction produces a rational root. Example: √(2/9). The denominator simplifies: √9=3. The numerator remains √2. Therefore: √(2/9)=√2/3. The Fraction Calculator can help when fraction arithmetic is the difficult part of a root problem.
Decimals can also have exact square roots. Example: √2.25. Because 1.5×1.5=2.25, √2.25=1.5. Another example: √0.0001=0.01 because 0.01²=0.0001. Place value matters when working with roots of decimals.
Suppose you want √50. Find nearby perfect squares. 49<50<64. Therefore: 7<√50<8. Because 50 is very close to 49, √50 should be only a little more than 7. The exact simplified radical is 5√2. Decimal: about 7.0711. This perfect-square bounding method is a useful quick estimate.
Before electronic calculators, several hand methods were used to calculate square roots. One method extracts the square root digit by digit in a process that has similarities to long division. At a high level, the method groups digits into pairs, determines one root digit at a time, subtracts the contribution of the partial square, and continues to additional digit pairs for greater precision. Because this is a longer algorithm than most people need for everyday calculations, this page does not turn into pages of mechanical arithmetic. Instead, here is a concise explanation and an optional video walkthrough. The educational YouTube video, watch a visual square-root algorithm walkthrough, demonstrates a digit-by-digit hand algorithm and explains the procedure visually.
Watch: How to Calculate the Square Root of Any Number, Digit by Digit MethodThe square-root function is y=√x. For real-number outputs, its domain is x≥0. Some easy points are: x=0→y=0, x=1→y=1, x=4→y=2, x=9→y=3, x=16→y=4, x=25→y=5. The graph starts at (0,0), increases as x increases, and gradually becomes flatter. It does not extend into negative x-values when working only with real square roots. The square-root function is related to y=x². If y=x² is restricted to x≥0, then y=√x is its inverse function.
Square roots appear whenever a squared quantity must be converted back into a length. One simple example is finding the side length of a square from its area. Suppose a square has area 144 square units. If the side is s, then s²=144. Take the square root: s=√144. Therefore: s=12. The Area Calculator can help with broader two-dimensional area problems. Square roots also appear in the Pythagorean theorem. If a right triangle has legs 3 and 4, then c²=3²+4². c²=25. Therefore: c=√25=5. The Pythagorean Theorem Calculator can show that process in detail.
Square roots appear when equations contain squared variables. Example: x²=49. Taking square roots gives: x=±7. Square roots also appear in the Quadratic Formula. The discriminant b²-4ac appears inside a square root. The Quadratic Formula Calculator can show how the radical determines whether a quadratic has two real roots, one repeated real root, or two complex roots.
Standard deviation is calculated as the square root of variance. Variance uses squared distances from the mean. Taking the square root returns the spread measurement to the original units of the data. For detailed sample and population calculations, see the Standard Deviation Calculator. This connection is a good example of why square roots are useful: squaring can make calculations mathematically convenient, and square roots can later return the value to the original scale.
A power moves forward: 5²=25. A root reverses that relationship: √25=5. Similarly: 3³=27. And: ∛27=3. The Scientific Calculator can help when roots are part of broader calculations involving powers, logarithms, or trigonometric functions.
Some positive integers are perfect squares. Their square roots are integers. Examples: √4=2, √9=3, √16=4. Other integers do not have rational square roots. For example: √2 cannot be written exactly as a fraction of two integers. Its decimal expansion never terminates and never becomes repeating. Therefore: √2 is irrational. The same is true for many roots such as √3 and √5. The fact that a decimal calculator displays only a limited number of digits does not mean the exact root ends there.
Consider √12. Simplify: √12=2√3. That is the exact form. Decimal: ≈3.4641. That is an approximation. Exact forms are useful in algebra because they preserve the precise value. Decimals are useful when you need measurements, estimates, comparisons, or a practical numerical answer. A strong root calculator should provide both whenever practical.
The principal radical √25 equals 5. The equation x²=25 has solutions ±5.
√16 is not 16÷2. It is the value whose square is 16. Answer: 4.
√72 can be simplified to 6√2.
Exact radical form can be more useful than a rounded decimal.
Only perfect squares have integer square roots.
A negative number does not have a real square root. Complex numbers are required.
Odd roots can be real for negative inputs. For example: ∛(-8)=-2.
√x means a second root. ∛x means a third root. 4√x means a fourth root.
The real square-root function y=√x has x≥0.
Keep exact values until the final decimal is needed.
For a perfect square: Step 1, identify the number (e.g. 144). Step 2, ask which nonnegative number times itself equals 144: 12×12=144. Step 3, therefore √144=12.
For a non-perfect square: Step 1, look for a perfect-square factor (e.g. √72). Step 2, factor: 72=36×2. Step 3, split the radical: √72=√36√2. Step 4, simplify: 6√2. Step 5, approximate only if needed: ≈8.4853.
Do not leave knowing only "√144=12." Leave knowing: I understand what a square root means. I know what the radical symbol represents. I understand what the radicand is. I understand the principal square root. I know why √25 is 5 while x²=25 has two real solutions. I know what a perfect square is. I understand why √2, √3, and √5 are irrational. I know how to simplify radicals. I know the difference between exact radical form and decimal approximation. I understand cube roots. I know negative numbers can have real odd roots. I understand why negative numbers do not have real even roots. I know what an nth root is. I understand roots as fractional exponents. I can work with roots of fractions and decimals. I know how to estimate a square root using nearby perfect squares. I understand that square roots can also be calculated by hand using digit-by-digit algorithms. I understand the square-root curve y=√x. I know square roots appear in area, the Pythagorean theorem, the Quadratic Formula, and standard deviation. And most importantly: I can see why the square root calculator produced the answer. That is what the CalculateThisWay Square Root Calculator should help someone understand.
| What You Need | Formula |
|---|---|
| Square Root | √x = x(1/2) |
| Cube Root | ∛x = x(1/3) |
| Nth Root | n√x = x(1/n) |
| Perfect Square | n² |
| Square Root Check | (√x)² = x for x≥0 |
| Root of a Fraction | √(a/b) = √a/√b when defined |
| Product Property | √(ab) = √a√b for nonnegative real a,b |
| Quotient Property | √(a/b) = √a/√b when b>0 |
| Inverse of Squaring | √(x²) = |x| |
| Square Area Side | s = √A |
| Cube Side From Volume | s = ∛V |
| n | n² | n | n² |
|---|
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Open Calculator ›What is a square root?
A square root of a number is the value that, multiplied by itself, produces that number. The square root of 25 is 5, because 5×5=25.
What does the square root symbol mean?
The radical symbol √ asks for the principal (nonnegative) square root of the radicand written underneath it.
What is a radicand?
The radicand is the number written inside a radical symbol, the number whose root is being taken. In √25, 25 is the radicand.
What is the principal square root?
It is the nonnegative square root of a nonnegative number. The radical symbol always refers to this principal root, so √36=6, not -6.
Is the square root of 25 5 or ±5?
√25 is 5 only. The ±5 answer applies to solving x²=25, which is a different question.
Why does x²=25 have two answers?
Because both 5²=25 and (-5)²=25 are true, so x=±5.
What is the square root of 2?
The exact value is √2, which cannot be simplified further. Its decimal approximation is about 1.41421356, and it is irrational.
What is the square root of 3?
The exact value is √3. Its decimal approximation is about 1.73205081.
What is the square root of 4?
√4=2, because 2×2=4. Four is a perfect square.
What is the square root of 5?
The exact value is √5. Its decimal approximation is about 2.23606798.
What is the square root of 16?
√16=4, because 4×4=16.
What is the square root of 100?
√100=10, because 10×10=100.
What is the square root of 144?
√144=12, because 12×12=144.
What is a perfect square?
A number that can be written as an integer squared, such as 1, 4, 9, 16, 25, and 36. Perfect squares have whole-number square roots.
How do I know if a number is a perfect square?
Take its square root. If the result is a whole number, it is a perfect square; if not, it is not.
How do you simplify a square root?
Find the largest perfect-square factor of the radicand, split the radical into a product, and take the square root of the perfect-square part.
How do I simplify √72?
72=36×2, so √72=√36×√2=6√2, approximately 8.4853.
Can a square root be irrational?
Yes. The square root of any positive integer that is not a perfect square is irrational.
Why is √2 irrational?
It cannot be written exactly as a fraction of two integers, and its decimal expansion never terminates or repeats.
What is the difference between exact and decimal square roots?
The exact answer, such as √2 or 6√2, preserves the precise value. A decimal, such as 8.4853, is a rounded approximation.
What is a cube root?
A value that, multiplied by itself three times, produces the original number. ∛27=3, because 3×3×3=27.
Can the cube root of a negative number be negative?
Yes. ∛(-8)=-2, because (-2)×(-2)×(-2)=-8.
Can I take the square root of a negative number?
Not as a real number. No real number squared produces a negative result. Representing it requires complex numbers.
What is an imaginary square root?
The square root of a negative number expressed using i, where i²=-1. For example, √(-16) has principal value 4i.
What is an nth root?
A value that, raised to the power n, equals x. A square root is an nth root with n=2.
What is a fourth root?
An nth root where n=4. 4√16=2, because 24=16.
What does n√x mean?
It means the value that, raised to the power n, equals x.
How are roots related to exponents?
n√x=x(1/n). Roots and fractional exponents describe the same operation.
Is √x the same as x(1/2)?
Yes, they are identical.
What is √(x²)?
√(x²)=|x|, the absolute value of x, not simply x.
Why is √(x²)=|x|?
Because the principal square root is always nonnegative, so when x is negative, the result must still be nonnegative, which is |x|.
How do I find the square root of a fraction?
Take the square root of the numerator and denominator separately: √(9/16)=√9/√16=3/4.
How do I find the square root of a decimal?
Find the value that, squared, returns the decimal. √2.25=1.5, because 1.5×1.5=2.25.
How do I estimate a square root?
Find the two nearest perfect squares. Since 49<50<64, √50 is between 7 and 8.
How do you calculate a square root without a calculator?
You can estimate with nearby perfect squares, or use a digit-by-digit hand algorithm similar in structure to long division.
What is the square root curve?
The graph of y=√x, which starts at the origin, exists only for x≥0, and increases more slowly as x grows.
What is the domain of y=√x?
x≥0, since the real square-root function is not defined for negative inputs.
What is the range of y=√x?
y≥0, since the principal square root is always nonnegative.
How do square roots appear in the Pythagorean theorem?
Finding the hypotenuse requires taking the square root of the sum of the squared legs, such as √25=5 for a 3-4-5 triangle.
How do square roots appear in the Quadratic Formula?
The formula includes √(b²-4ac), which determines whether a quadratic has two real solutions, one repeated real solution, or two complex solutions.
Why is standard deviation a square root?
Standard deviation is the square root of variance, converting squared units back into the original units of the data.
What is the difference between roots and powers?
A power multiplies a base by itself repeatedly, such as 5²=25. A root reverses that operation, such as √25=5.
Square-root results use the principal nonnegative real root for nonnegative inputs. Cube roots and other odd nth roots may accept negative real inputs, since odd roots of negative real numbers are real. Even roots of negative inputs have no real result in this calculator's real-number mode; a square root is the only case where an optional complex principal root (such as 4i for the square root of -16) is also shown. Exact radicals are simplified using perfect-power factors where practical, and exact fractions are kept and reduced rather than converted to decimals prematurely. Decimal results are approximations whenever the exact root is irrational. Displayed decimals may be rounded for readability, while calculations use greater internal precision; use the decimal-places control to see additional digits. Perfect-square checks determine whether the principal square root of a whole number is itself a whole number. Nth-root calculations use mathematically valid real-number domain rules: an even root index requires a nonnegative input, and an odd root index accepts any real input. Complex results are clearly labeled and distinguished from real results whenever they are shown. This calculator does not claim infinite numerical precision.
Last reviewed: September 24, 2026