Enter a number and an index into the root calculator above to get your answer instantly. Whether you need a square root, cube root, or any nth root of a number, this free tool returns the result with step-by-step detail.
A root answers a simple question: what number, multiplied by itself a certain number of times, produces the value you started with? The square root of 9 is 3, because 3 × 3 = 9. The cube root of 27 is 3, because 3 × 3 × 3 = 27. This calculator handles all of those cases and more.
Square Root Calculator
The square root is the most common root in everyday math. It finds the number that, when multiplied by itself once, equals your input. The symbol √ (called a radical sign) represents this operation.
For example, √49 = 7. That's because 7 × 7 = 49.
Use the calculator above by entering your number and leaving the root index set to 2. The tool returns both the exact value (when the number is a perfect square) and a decimal approximation for numbers that aren't.
What is a square root? A square root of a number x is a value that, when squared, gives x. Every positive number has two square roots: one positive and one negative. By convention, the radical symbol √ refers to the positive root only. So √25 = 5, even though (−5) × (−5) also equals 25.
How to Calculate the Square Root of a Number
Calculating square roots by hand is straightforward for perfect squares. For other numbers, you can estimate or use a systematic method.
Finding the Square Root by Method
There are several ways to calculate square roots without a calculator:
- Prime factorization. Break the number into prime factors. Pair them up. One factor from each pair comes outside the radical. For example, 36 = 2 × 2 × 3 × 3, so √36 = 2 × 3 = 6.
- Estimation and refinement. Pick a guess, divide your number by the guess, then average the guess with the quotient. Repeat until the answer stabilizes. This is sometimes called the Babylonian method.
- Long division method. Group digits in pairs from the decimal point outward, then work through them step by step, similar to long division. This gives precise decimal results.
For most practical purposes, the calculator on this page handles these steps for you instantly.
Square Root of 25 and Other Common Numbers
Memorizing common square roots saves time in schoolwork and quick estimates. Here are values worth knowing:
- √4 = 2
- √9 = 3
- √16 = 4
- √25 = 5
- √36 = 6
- √49 = 7
- √64 = 8
- √81 = 9
- √100 = 10
- √144 = 12
These are all perfect squares, meaning their square roots are whole integers. When a number is a perfect square, the result is exact. When it is not (like √2 or √7), the decimal goes on forever without repeating.
Nth Root Calculator
Sometimes you need more than a square root. The nth root generalizes the concept: the nth root of a number x is the value r such that r raised to the nth power equals x.
Mathematically: ∛x (cube root), ⁴√x (fourth root), or more generally, the nth root of x.
Set the index in the calculator to any positive integer (2 for square root, 3 for cube root, 4 for fourth root, and so on) and enter your number.
Finding the Nth Root of a Number
To find the nth root of a number manually, you can use this relationship:
The nth root of x = x^(1/n)
So the 5th root of 32 is 32^(1/5) = 2, because 2⁵ = 32.
For numbers that don't resolve to clean integers, the calculator provides a decimal approximation. This estimate is reliable for planning and homework. For applications requiring certified precision (engineering tolerances, for example), verify results with professional tools.
Cube Root Calculator
The cube root is the second most common root after the square root. It finds the number that, when multiplied by itself three times, equals the input.
Key difference from square roots: cube roots work on negative numbers too. The cube root of −8 is −2, because (−2) × (−2) × (−2) = −8.
Common cube roots to know:
- ∛1 = 1
- ∛8 = 2
- ∛27 = 3
- ∛64 = 4
- ∛125 = 5
- ∛1000 = 10
Root of a Number in Math
What Is a Root in Math?
The word "root" carries two distinct meanings in math, and they often get mixed up.
Root as a radical (nth root). This is what the calculator on this page computes. You provide a number and ask: what base value, raised to a given power, produces this number? The radicand is the number under the radical sign. The index tells you the degree of the root.
Root of an equation. Here, "root" means a solution, specifically the value of x that makes an equation equal zero. For instance, the roots of x² − 9 = 0 are x = 3 and x = −3. These are the points where a graph crosses the x-axis.
Both meanings connect to the same underlying math, but they answer different questions. This calculator focuses on the first meaning: computing the nth root of a given number.
Square Number and Root
A square number (also called a perfect square) is any integer multiplied by itself. The numbers 1, 4, 9, 16, 25, 36, and so on are all perfect squares.
The square root reverses this process. If you know the square number, the root tells you what integer produced it. This relationship is foundational in geometry (finding side lengths from areas), physics, and statistics.
How do you find complex roots? When you take an even root of a negative number (like √−4), there is no real number answer. The result involves imaginary numbers. √−4 = 2i, where i represents √−1. This calculator works with real number inputs. For complex roots, a specialized math tool is a better fit.
Finding Roots of a Polynomial Equation
How to Find the Roots of a Polynomial
Finding the roots of a polynomial means solving for the values of x where the polynomial equals zero. A polynomial like x² − 5x + 6 = 0 has roots at x = 2 and x = 3.
The approach depends on the polynomial's degree:
- Degree 1 (linear). Solve directly. Example: 2x − 6 = 0 → x = 3.
- Degree 2 (quadratic). Factor, complete the square, or use the quadratic formula.
- Degree 3 and higher. Try the rational root theorem, synthetic division, or numerical methods.
The radical calculator above handles nth root computation, which is one step in many of these solving methods. For full polynomial solving with multiple roots, a dedicated polynomial roots calculator is more appropriate.
Method for Solving a Polynomial Equation
Here is a practical method for lower-degree polynomials:
- Set the polynomial equal to zero. Move all terms to one side.
- Look for common factors. Factor out any shared terms first.
- Factor the expression. For quadratics, find two numbers that multiply to the constant term and add to the middle coefficient.
- Set each factor equal to zero. Solve for x in each case.
- Check your solutions. Substitute back into the original equation.
For a quadratic ax² + bx + c = 0, the quadratic formula always works:
x = (−b ± √(b² − 4ac)) / 2a
Notice the square root inside the formula. When b² − 4ac is negative, the equation has no real roots, only complex ones.
Root Calculator for Scientific Math
Using a Scientific Calculator to Find Roots
Most scientific calculators include a dedicated root function, but the button labels vary across brands. Here is how to find roots on common calculator types:
- Square root. Press the √ button, then enter the number. On some models, enter the number first.
- Cube root. Look for a ∛ button or use the nth root function with 3.
- Nth root. Find the key labeled ⁿ√x or x^(1/n). Enter the index, press the function key, then enter the radicand.
If your physical calculator doesn't have an nth root button, use the exponent key instead. Enter the number, press the exponent key (often labeled ^), then type (1/n) where n is your root index. For example, the 4th root of 81: type 81^(1/4) = 3.
The online root calculator at the top of this page works the same way, with no special button hunting required.
Real Number Roots and Root Calculator Limitations
Every calculator has boundaries. Here is what to keep in mind:
Even roots of negative numbers have no real solution. √−16 or the 4th root of −25 will produce an error or an imaginary number. The calculator works within the real number system. If you need complex solutions, note that the result involves i (the imaginary unit).
Odd roots of negative numbers are valid. The cube root of −27 is −3. The 5th root of −32 is −2. Odd-index roots always return a real result, whether the input is positive or negative.
Decimal precision. Most roots of non-perfect powers produce irrational numbers, meaning the decimal never terminates or repeats. The calculator displays a rounded approximation. This is accurate enough for homework, estimation, and general planning. It is not a substitute for a computer algebra system when exact symbolic answers matter.
Very large or very small inputs. Extremely large numbers (hundreds of digits) may exceed the precision of standard floating-point math. For typical use, the values you enter will compute correctly.
This tool provides estimates and educational support. For professional engineering, financial, or scientific applications where certified precision is required, verify results with industry-standard software.