Root Calculator — Free Online Nth Root Calculator

How to Use the Root Calculator

1. Enter the radicand: Type the number whose root you want to find in the Radicand field. This is the value inside the radical sign. It can be any real number — enter 144 for a perfect square, 2 for an irrational root, or even a negative number if using an odd root index.
2. Set the root index: Enter the root index (n) to specify which root to calculate. Use 2 for square root (the default), 3 for cube root, 4 for fourth root, and so on. The minimum index is 1. Fractional indices are not supported — use our exponent calculator for fractional powers.
3. Review the results: The result panel instantly shows the principal root value, a verification calculation (the result raised back to the power of n, which should equal your original radicand), and comparison values for the square root, cube root, and fourth root of the same number. Everything updates in real time.

The calculator handles edge cases automatically — even roots of negative numbers show "undefined" (since they produce complex numbers), and the zeroth root is flagged as undefined. The verification value lets you confirm the answer is correct by checking that it powers back to the original number.

Root Formulas and Mathematical Foundation

Variables Explained

• Radicand (x): The number under the radical sign. This is the value whose root you want to find. For square roots, x must be non-negative to produce a real result. For odd roots, x can be any real number.
• Root Index (n): The degree of the root. An index of 2 gives a square root, 3 gives a cube root, and so on. The index must be a positive integer (1 or greater). Higher indices produce values closer to 1 for radicands greater than 1.
• Result: The principal nth root of the radicand. This is the non-negative real number y such that y^n = x (for positive radicands). For negative radicands with odd indices, the result is the unique real root.

Step-by-Step Example

Calculate the cube root of 216:

1. Set radicand = 216, root index = 3
2. Factor 216: 216 = 6 x 6 x 6 = 6^3
3. The cube root of 216 = 216^(1/3) = (6^3)^(1/3) = 6
4. Verify: 6^3 = 6 x 6 x 6 = 216 (correct)

When a number is a perfect cube, the cube root is a whole number. If it is not a perfect cube, like 200, the cube root is irrational: the cube root of 200 is approximately 5.8480.

Practical Examples

Example 1: Amanda Calculates a Room Diagonal

Amanda is buying a new TV and needs to measure the diagonal distance of the wall space. The wall section is 48 inches wide and 36 inches tall. Using the Pythagorean theorem, the diagonal is the square root of (48^2 + 36^2):

• 48^2 = 2,304
• 36^2 = 1,296
• Sum = 2,304 + 1,296 = 3,600
• Square root of 3,600 = 60 inches

Amanda can fit a TV up to 60 inches diagonally in this space. The 36-48-60 combination is actually a multiple of the classic 3-4-5 right triangle (each multiplied by 12).

Example 2: Kevin Determines a Storage Container Size

Kevin needs a cubic storage container that holds exactly 2,744 cubic inches. He needs to know the side length. Since the container is a cube, each side is the cube root of the volume:

• Cube root of 2,744 = 2,744^(1/3)
• Factor: 2,744 = 14 x 14 x 14 = 14^3
• Cube root of 2,744 = 14 inches per side

Kevin needs a container that is 14 inches on each side. This cube root relationship between volume and side length is fundamental in engineering, architecture, and packaging design.

Example 3: Sophie Finds the Standard Deviation

Sophie is analyzing test scores and calculated the variance of her data set as 196. The standard deviation is the square root of the variance:

• Variance = 196
• Standard deviation = square root of 196 = 14

The standard deviation of 14 points tells Sophie that most scores in her data set fall within 14 points of the mean. For comprehensive statistical analysis, try our standard deviation calculator.

Example 4: Nathan Computes Average Annual Return

Nathan invested $5,000 that grew to $8,000 over 6 years. To find the compound annual growth rate (CAGR), he needs the 6th root of (8,000 / 5,000):

• Growth factor: 8,000 / 5,000 = 1.6
• CAGR = (1.6)^(1/6) - 1
• 6th root of 1.6 = 1.0814
• CAGR = 0.0814 = 8.14% per year

Nathan earned an average annual return of about 8.14%. This nth-root approach is how financial analysts calculate annualized returns from total growth over multiple years.

Common Roots Reference Table

Tips and Complete Guide

Estimating Roots Without a Calculator

You can estimate square roots by finding the two perfect squares that bracket your number. To estimate the square root of 50: 49 = 7^2 and 64 = 8^2, so the square root of 50 is between 7 and 8, closer to 7 since 50 is closer to 49. A more precise estimate uses linear interpolation: 7 + (50 - 49)/(64 - 49) = 7 + 1/15 = approximately 7.07. The actual value is 7.0711. This technique is handy for quick mental math and for checking calculator results.

Roots in Geometry and Physics

Square roots are essential in geometry for calculating distances (Pythagorean theorem), diagonals (a square with side s has diagonal s times the square root of 2), and areas (a circle with area A has radius equal to the square root of A/pi). In physics, the period of a pendulum is proportional to the square root of its length, escape velocity involves the square root of 2GM/r, and the RMS (root mean square) voltage is calculated using square roots. Cube roots appear in volume-to-length conversions and in fluid dynamics.

Rationalizing the Denominator

In algebra, it is standard practice to rationalize denominators — removing radicals from the bottom of fractions. To rationalize 1/sqrt(3), multiply both numerator and denominator by sqrt(3): the result is sqrt(3)/3. For binomial denominators like 1/(1 + sqrt(2)), multiply by the conjugate (1 - sqrt(2))/(1 - sqrt(2)) to get (1 - sqrt(2))/(1 - 2) = sqrt(2) - 1. This technique simplifies expressions and makes them easier to compare and compute.

Historical Significance of Root Calculations

The discovery that the square root of 2 is irrational was one of the most important moments in mathematical history, attributed to the Pythagorean school around 500 BCE. This shattered the belief that all numbers could be expressed as ratios of whole numbers. The Babylonians used an iterative method to approximate square roots as early as 1800 BCE — a method essentially equivalent to Newton's method. Today, computers use optimized algorithms to calculate roots to billions of decimal places, but the fundamental mathematical principles remain unchanged.

Common Mistakes to Avoid

• Assuming sqrt(a + b) = sqrt(a) + sqrt(b): This is false. The square root of a sum is NOT the sum of the square roots. For example, sqrt(9 + 16) = sqrt(25) = 5, but sqrt(9) + sqrt(16) = 3 + 4 = 7. Always compute the sum first, then take the root.
• Forgetting that square roots have two values: While calculators return the principal (positive) root, the equation x^2 = 25 has two solutions: x = 5 and x = -5. When solving equations, always consider both roots unless context restricts to positive values.
• Trying to take even roots of negative numbers: The square root of -4 is not -2 (since (-2)^2 = +4, not -4). Even roots of negative numbers require complex numbers. Only odd roots of negative numbers produce real results.
• Rounding too early in multi-step calculations: When a root appears in an intermediate step, carry as many decimal places as possible before rounding the final answer. Premature rounding can lead to significant errors, especially in chain calculations.
• Confusing root index with the result: The cube root of 8 is 2 (not 3). The root index tells you the degree of the root, not the answer. A higher root index produces a result closer to 1 for numbers greater than 1.

Frequently Asked Questions