2026-08-07 10 min

Root Calculator

Calculate square root, cube root, and nth root with step-by-step estimation. Free online root calculator with Babylonian and Newton iterations.

Calculator

Root Inputs

Radicand (a) Help

Number under the radical

Root degree (n) Help

2 = square, 3 = cube

Results update automatically

Results update live as you type

Status:a = 27, n = 3nth root = 3sqrt = 5.1961524227

Root Summary

Square root√27 = 5.1961524227

Cube root³√27 = 3

Degree-3 root3

Verification root^n27

Common Roots Table

n	√n	³√n
1	1	1
2	1.4142135624	1.2599210499
3	1.7320508076	1.4422495703
4	2	1.587401052
5	2.2360679775	1.7099759467
6	2.4494897428	1.8171205928
7	2.6457513111	1.9129311828
8	2.8284271247	2
9	3	2.0800838231
10	3.1622776602	2.15443469

Estimation Trace

Babylonian for √27: b0 = 13.5000, b1 = 7.75 → converges to 5.1961524227

Newton for degree-3: x1 = ((n-1)x + a/x^(n-1))/n → converges to 3

Results

Optimal

nth Root

Degree-3 root of 27

Square Root

5.1961524227

Index 2 root of the radicand

Cube Root

Index 3 root of the radicand

Verification (root^n)

Raising the nth root back to the degree should return the radicand

Radicand Squared

729

a squared for scale comparison

Radicand Cubed

19683

a cubed for scale comparison

Root Degree (n)

Index of the computed nth root

Real principal roots computed via fractional exponents; verification row powers the root back to the radicand. Non-real even roots of negatives show N/A.

Visualizations

Square vs Cube vs nth Root

5.2sqrt

3cbrt

3n=3

Root Magnitude Split

nth root11%

Remainder to radicand89%

Verification Closeness

Match100%

nth Root vs Radicand

Root3

Formula

The nth root of a is the number x with x^n = a. Square roots use n = 2, cube roots n = 3, and general roots use fractional exponents a^(1/n).

nth root: x = a^(1/n); x^n = a

Alternative Form

sqrt(a) = a^(1/2); cbrt(a) = a^(1/3)

Variables

Symbol	Name	Unit	Description
a	Radicand	number	Value under the radical
n	Degree	count	Index of the root
x	Root	number	a^(1/n)
x^n	Verification	number	Should equal a

Explanation

Roots undo exponentiation. The curve y = x^(1/n) rises steeply then flattens; the marked point shows where the input radicand lands on that curve.

Derivation

x = nthroot(a, n) = a^(1/n). Check: x^n = (a^(1/n))^n = a. Babylonian step for n=2: b_new = (b + a/b)/2. Newton step for general n: x_new = ((n-1)x + a/x^(n-1))/n.

Step-by-Step Calculation

Follow each step to understand how the result is calculated from your inputs.

Step 1

Read the radicand a and the degree n.

a, n given

radicand:27

degree:3

Result: Compute degree-3 root of 27

Step 2

Compute the square root (n = 2) for reference.

sqrt(a) = a^(1/2)

a:27

Intermediate: 27^(1/2)

Result: sqrt = 5.1961524227

Step 3

Compute the cube root (n = 3) for reference.

cbrt(a) = a^(1/3)

a:27

Intermediate: 27^(1/3)

Result: cbrt = 3

Step 4

Compute the general nth root via exponentiation.

nth = a^(1/n)

a:27

n:3

Intermediate: 27^(1/3)

Result: nth root = 3

Step 5

Verify by raising the root back to the degree.

check = nth^n

nth:3

n:3

Intermediate: 3^3

Result: check = 27 (target 27)

Step 6

Refine with one Babylonian/Newton iteration if needed.

x1 = ((n-1)x + a/x^(n-1))/n

a:27

n:3

Intermediate: Iteration converges in a few steps

Result: Final approximation = 3

Final Answer

Worked Examples

See how the calculator works with different scenarios.

Scenario

Perfect square radicand with degree 2.

Inputs

radicand:100

degree:2

Calculation

sqrt(100) = 100^(1/2) = 10 check: 10^2 = 100

Result

nth root = 10, verified exactly

Interpretation

Perfect squares return integers; the curve point sits exactly on the grid.

Understanding Your Result

Your Result: nth Root: 3

Typical range: Square roots of areas give side lengths; cube roots of volumes give edge lengths; higher roots shrink values above 1 and grow values between 0 and 1.

What This Means

The nth root result is the number that, raised to the degree, reproduces the radicand. Square and cube roots are shown alongside for reference.

Why It Matters

Roots size distances, volumes, standard deviations, and financial volatilities. Knowing which degree applies prevents order-of-magnitude errors.

Professional Interpretation

Check the verification row: if root^n differs from the radicand beyond rounding, the degree or sign handling is wrong (often a negative under an even root).

Recommendations

•Verify root^n equals the radicand
•Keep even roots non-negative
•Leave surds in radical form for exact work
•Use Newton iteration for high degrees
•Convert to fractional exponents for calculus

Educational Content

Introduction

The root calculator computes square, cube, and nth roots from a radicand and a degree, then verifies the answer by powering it back up.

Concept Explanation

x = nthroot(a, n) exactly when x^n = a. The principal real root is non-negative for even n and follows the sign of a for odd n.

Applications

▸Pythagorean distances and hypotenuses

▸Cube edge from volume via cube roots

▸Standard deviation as a square root of variance

▸RMS voltage and signal power

▸Scaling laws with fourth and higher roots

▸Solving quadratic equations

Benefits

•Exact verification via powering back
•Handles square, cube, and general degrees
•Shows estimation iterations
•Flags non-real even roots of negatives
•Teaches exponent-root duality

Limitations

•Real roots only (no complex output)
•Even roots of negatives are N/A
•Floating rounding near 1e-10
•Degree clamped to 1-20

Best Practices

•Check root^n against the radicand
•Simplify perfect-power factors first
•Use surd form for exact geometry
•Pick odd degrees for negative inputs
•Round only at the final step

Common Misconceptions

Myth: sqrt(a + b) equals sqrt(a) + sqrt(b).

Truth: Addition does not distribute inside radicals; compute the sum first, then take the root.

Myth: Every number has a real square root.

Truth: Negative numbers have no real square root — only odd-index roots of negatives are real.

Myth: Roots and exponents are unrelated.

Truth: They are inverses: nthroot(a, n) = a^(1/n), which is why fractional exponents appear in calculus.

Myth: Decimal approximations are always better.

Truth: Surds like sqrt(2) are exact in radical form; decimals truncate an infinite expansion.

Practical Tips

•Verify root^n equals the radicand
•Use degree 2 for areas, 3 for volumes
•Factor out perfect powers before rooting
•Keep surds exact until the last step
•Use odd degrees for negative inputs
•Iterate Babylonian/Newton twice for full precision
•Convert to a^(1/n) for derivatives and integrals

Common Mistakes

Adding inside the radical term-by-term

Sum first, then root: sqrt(9 + 16) = sqrt(25) = 5, not 3 + 4 = 7.

Taking an even root of a negative

Switch to an odd degree or complex mode; real even roots need a non-negative radicand.

Forgetting extraneous solutions

After solving radical equations, substitute back — powering both sides can invent solutions.

Writing sqrt(x^2) as x

It equals |x|; the sign matters for negative x.

Rounding intermediate iterations

Carry full precision through Babylonian/Newton steps and round only the final display.

Assumptions

This calculator makes the following assumptions to provide estimates:

Real numbers only: No complex output; non-real cases display N/A.

Principal root: Even roots return the non-negative branch.

Integer degree: Degree is rounded to an integer between 1 and 20.

Finite inputs: Non-finite or empty inputs fall back to defaults.

Power verification: check = nth^degree should reproduce the radicand within rounding.

Limitations

Be aware of these limitations when using this calculator:

No complex roots: Imaginary results for negative even roots are not computed.

Floating precision: Irrational surds are shown to 10 decimals.

Degree clamp: Degrees above 20 are clamped to keep pow() stable.

No symbolic simplification: Outputs are numeric, not factored surd forms like 5*sqrt(2).

Frequently Asked Questions

Answers to common questions about this calculator.

What is a root in mathematics?

A root (or radical) of a number a is a value x such that x^n = a, where n is the index or degree of the root. The most common roots are the square root (n = 2) and the cube root (n = 3). The nth root of a is written as the nth radical of a. Roots are the inverse operation of exponentiation.

What is the difference between a square root and a cube root?

A square root (index 2) finds a value that when multiplied by itself gives the original number. A cube root (index 3) finds a value that when multiplied by itself three times gives the original number. For example, the square root of 64 is 8 (8 x 8 = 64), while the cube root of 64 is 4 (4 x 4 x 4 = 64). Cube roots can be taken of negative numbers, while real square roots of negative numbers do not exist.

Can you take the square root of a negative number?

The square root of a negative number is not a real number — it is an imaginary number. For example, the square root of -1 is i, the imaginary unit. This calculator only computes real-number square roots, so entering a negative number for the square root shows N/A. Use a complex number calculator for square roots of negatives.

What is the nth root of a number?

The nth root of a number a is a number x such that x^n = a. The index n determines the degree. For n = 2 it is the square root, n = 3 the cube root, n = 4 the fourth root, and so on. The 4th root of 81 is 3 because 3^4 = 81. When n is even the radicand must be non-negative for a real result; when n is odd it can be any real number.

What is the difference between a radical and an exponent?

A radical (root) and an exponent are inverse operations. The nth root of a equals a^(1/n). For example, sqrt(9) = 9^(1/2) = 3 and cbrt(8) = 8^(1/3) = 2. Radical notation is often preferred for exact values, while exponential notation is more convenient for algebraic manipulation.

How do you simplify radicals?

To simplify a radical, factor out perfect powers from the radicand. For square roots look for perfect square factors: sqrt(50) = sqrt(25 x 2) = 5*sqrt(2). For cube roots look for perfect cube factors: cbrt(40) = cbrt(8 x 5) = 2*cbrt(5). Simplified radicals have no perfect power factors left inside.

What is the Babylonian method for square roots?

The Babylonian method (Herons method) is an iterative algorithm: start with a guess b, compute c = a / b, then update b = (b + c) / 2 and repeat until b and c agree. Each iteration roughly doubles the number of correct digits. For sqrt(27) with b = 5: c = 5.4, new b = 5.2, and a few more steps converge to about 5.19615.

What is Newtons method for nth roots?

Newtons method gives the iteration x(k+1) = (1/n) x ((n-1) x x(k) + a / x(k)^(n-1)). Start with a guess x0 and iterate until convergence. It converges quickly, typically needing only a few iterations for good accuracy, and handles any integer degree n.

What is a perfect square?

A perfect square is an integer that is the square of another integer: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, and so on. The square root of a perfect square is always an integer. Perfect squares are commonly used when simplifying radicals.

What is a perfect cube?

A perfect cube is an integer that is the cube of another integer: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000. The cube root of a perfect cube is always an integer. Perfect cubes help simplify cube root expressions.

How is the square root used in the Pythagorean theorem?

The Pythagorean theorem (a^2 + b^2 = c^2) uses square roots directly. The hypotenuse is c = sqrt(a^2 + b^2). For legs 3 and 4: c = sqrt(9 + 16) = sqrt(25) = 5. To find a leg: a = sqrt(c^2 - b^2).

What are surds in mathematics?

A surd is a radical expression that cannot be simplified to remove the radical, such as sqrt(2), sqrt(3), or sqrt(5). Surds are irrational numbers with non-repeating infinite decimal expansions and are often left in radical form for exact expressions.

How do you solve radical equations?

Isolate the radical on one side and raise both sides to the power of the index. For example, sqrt(x + 3) = 5 becomes x + 3 = 25, so x = 22. Always check solutions by substituting back, as extraneous solutions may arise from the powering step.

What are the most common mistakes with radicals?

Common mistakes include distributing addition inside a radical (sqrt(a + b) is not sqrt(a) + sqrt(b)), forgetting that sqrt(x^2) = |x|, mishandling negatives under even-index radicals, skipping the extraneous-solution check, and cancelling terms inside a radical incorrectly.