Calculate square roots, cube roots, and general nth roots instantly. Enter any number, get your result, see whether it is a perfect square or cube, and follow the manual estimation steps.
Enter a number to calculate its square root
Enter a number to calculate its cube root
Enter a number and root index to calculate
This tool handles square roots, cube roots, and any nth root in three separate tabs. Follow the steps below to get your result in seconds.
Select the tab that matches your calculation: Square Root for √x, Cube Root for ³√x, or General Root for any nth root including 4th, 5th, and beyond.
Type any positive number into the input field. Decimal values such as 12.5 or 0.04 are fully supported. For cube roots and odd-indexed general roots, negative numbers are also accepted.
In the General Root tab, enter the index n (2, 3, 4, 5, etc.). The default is 2, giving you the square root. Odd indices work for negative bases; even indices require a positive base in real numbers.
The result panel shows your root value, whether the base is a perfect square or cube, and a verification check confirming that (result)n equals your original number — so you can trust the answer.
In mathematics, the general root — or the nth root of a number a — is another number b that when multiplied by itself n times equals a.
n√a = b
bn = a
After clicking Calculate, the result panel gives you four pieces of information:
A quick example: entering 625 in the Square Root tab returns 25, type "Perfect Square," and verification 25² = 625. Entering 50 returns 7.0710678118, type "Irrational," and verification 7.071² ≈ 50.
Every root can be expressed as a fractional exponent. The nth root of a equals a raised to the power of 1/n:
n√a = a1/n
This equivalence means that √a = a0.5, ³√a = a0.333…, and so on. On any scientific calculator with an exponent key, you can compute any root using this form. On this page, the General Root tab handles all of that automatically.
When you need to estimate roots by hand, there is a simple iterative method — sometimes called the Babylonian or Newton–Raphson method — that converges quickly with just a few arithmetic steps.
Calculating square roots manually is intensive, but you can estimate them by repeating a simple divide-and-average process.
To calculate √a:
Guess: 5.125
27 ÷ 5.125 = 5.268
(5.125 + 5.268) / 2 = 5.197
27 ÷ 5.197 = 5.195
(5.195 + 5.197) / 2 = 5.196
27 ÷ 5.196 = 5.196 — converged. √27 ≈ 5.196
Higher roots use a similar method with a small change to account for the root index.
To calculate n√a:
Guess: 1.432
15 ÷ 1.4327 = 1.405
(1.432 × 7 + 1.405) / 8 = 1.388
15 ÷ 1.3887 = 1.403
(1.403 × 7 + 1.388) / 8 = 1.402
Continuing further still rounds to 1.403. 8√15 ≈ 1.403
Root calculations appear across geometry, finance, physics, and statistics. Here are six practical scenarios with real numbers you can verify in the calculator above.
If a square room has a floor area of 625 sq ft, the side length is √625 = 25 ft. Enter 625 in the Square Root tab to confirm. The result shows "Perfect Square," meaning 25 × 25 = 625 exactly, with no rounding required.
If an investment grew from $10,000 to $14,641 over 4 years, the annual growth rate is the 4th root of (14,641 ÷ 10,000) minus 1. Enter 1.4641 in the General Root tab with root index 4: 4√1.4641 = 1.1, so the CAGR is 10%. This technique is used widely in US retirement and brokerage account analysis.
A circular pipe cross-section has an area of 78.54 cm². Since A = πr², the radius is r = √(A / π) = √(78.54 / 3.14159) = √25 = 5 cm. Enter 25 in the Square Root tab to get 5 instantly. Plumbers and HVAC engineers use this daily when sizing ductwork and conduit.
Standard deviation is the square root of variance. If a dataset has a variance of 36, the standard deviation is √36 = 6. This is a perfect square, so the result is a clean whole number. In practice, variance is rarely a perfect square, and the square root produces an irrational decimal result like 4.8989…
The time in seconds for an object to fall from rest through a height h (in feet) near Earth's surface is t = √(2h / 32). From 200 feet: t = √(400 / 32) = √12.5 ≈ 3.536 seconds. Enter 12.5 in the Square Root tab to get 3.5355339059. Physics students use this to calculate drop times and verify free-fall experiments.
By the Pythagorean theorem, the diagonal d of a rectangle with sides 9 ft and 12 ft is d = √(9² + 12²) = √(81 + 144) = √225 = 15 ft. This is a 3-4-5 Pythagorean triple scaled by 3. Builders use this technique to verify that room corners are square during framing.
Memorizing the most common perfect squares and cubes speeds up mental math, algebra simplification, and recognizing irrational results. If your input appears in the "Square" or "Cube" column below, the corresponding root is a whole number — and this calculator will confirm it with the "Perfect Square" or "Perfect Cube" label.
| n | Square (n²) | Square Root (√n²) | Cube (n³) | Cube Root (³√n³) |
|---|---|---|---|---|
| 1 | 1 | 1 | 1 | 1 |
| 2 | 4 | 2 | 8 | 2 |
| 3 | 9 | 3 | 27 | 3 |
| 4 | 16 | 4 | 64 | 4 |
| 5 | 25 | 5 | 125 | 5 |
| 6 | 36 | 6 | 216 | 6 |
| 7 | 49 | 7 | 343 | 7 |
| 8 | 64 | 8 | 512 | 8 |
| 9 | 81 | 9 | 729 | 9 |
| 10 | 100 | 10 | 1,000 | 10 |
| 11 | 121 | 11 | 1,331 | 11 |
| 12 | 144 | 12 | 1,728 | 12 |
| 13 | 169 | 13 | 2,197 | 13 |
| 14 | 196 | 14 | 2,744 | 14 |
| 15 | 225 | 15 | 3,375 | 15 |
| 20 | 400 | 20 | 8,000 | 20 |
| 25 | 625 | 25 | 15,625 | 25 |
Tip: if your input does not appear in the Square or Cube column, the calculator will label the result "Irrational" — a useful quick check when simplifying radical expressions or checking textbook answers.
Quick links to closely related math tools that pair well with root calculations.
Use a full scientific calculator for exponent form, powers, and multi-step expressions that include radicals and fractional exponents.
Add, subtract, multiply, and divide fractions — useful when simplifying radical expressions that involve fractional exponents like a1/n.
Calculate percentages, percentage change, and percentage of a total — pairs well with CAGR calculations that use nth roots.
Calculate weighted grades, final percentages, and GPA-style results when you need quick academic math beyond radicals.
Calculate days between dates, add or subtract days, and plan timelines accurately for study schedules, projects, or deadlines.
Estimate daily calorie needs for weight loss, maintenance, or gain using age, weight, height, and activity level.
Answers to the most common questions about square roots, cube roots, and nth roots — including edge cases, manual methods, and real-number limits.
The nth root of a number a is another number b that, when multiplied by itself n times, equals a. Written as n√a = b, this means bn = a. For example, the 4th root of 81 is 3, because 3 × 3 × 3 × 3 = 81. Square roots (n = 2) and cube roots (n = 3) are the most common types, but this calculator handles any index you enter.
Use the divide-and-average method. To find √50: start with a guess of 7 (since 7² = 49 is close). Divide 50 by 7 to get 7.143. Average your guess and the result: (7 + 7.143) / 2 = 7.071. Repeat: 50 ÷ 7.071 ≈ 7.071. You have converged. Each iteration roughly doubles the number of correct decimal places, so three or four rounds usually give you six-digit accuracy.
A square root (√) finds a number that, multiplied by itself twice, equals the original. A cube root (³√) finds a number multiplied by itself three times. For example, √25 = 5 because 5² = 25, while ³√27 = 3 because 3³ = 27. An important practical difference: cube roots work for negative numbers (³√(−8) = −2), whereas square roots of negative numbers are undefined in real numbers.
A perfect square is any non-negative integer whose square root is also a whole number. Common examples: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, and 225. Recognizing perfect squares lets you simplify radical expressions quickly — for instance, √72 = √(36 × 2) = 6√2 — and verify calculator results at a glance. The reference table above lists every perfect square up to 625.
A perfect cube is any integer whose cube root is a whole number. The most common perfect cubes to know: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1,000, 1,331, and 1,728. For example, 512 is a perfect cube because ³√512 = 8 and 8³ = 512. This calculator labels results as "Perfect Cube" automatically when your input falls into this category.
Not within the real number system. No real number squared produces a negative result, so √(−1) has no real-number solution. In complex mathematics, √(−1) is defined as i (the imaginary unit), and √(−a) = i√a for any positive a. This calculator operates in real numbers only, so entering a negative value in the Square Root tab shows an error. Odd-indexed roots (cube root, 5th root, etc.) do accept negative bases.
Roots and exponents are inverse operations. The nth root of a equals a raised to the power 1/n: n√a = a1/n. So √a = a0.5, ³√a = a0.333…, and √[4]{a} = a0.25. This is why scientific calculators can compute any root using the yx or ^ key: enter the base, press ^, then type 1 divided by n. This calculator handles that conversion automatically in the General Root tab.
An irrational number cannot be expressed as a fraction of two integers. Its decimal expansion continues infinitely without any repeating pattern. √2 = 1.41421356…, √3 = 1.73205080…, and √5 = 2.23606797… are classic examples. Any positive integer that is not a perfect square will produce an irrational square root. This calculator shows up to 10 decimal places and labels the result "Irrational" so you know the displayed value is a rounded approximation.
The 4th root of a number a is the value b where b × b × b × b = a. Using this calculator, enter your number in the General Root tab and set the root index to 4, then click Calculate. Alternatively, use the exponent relationship: 4√a = a0.25. For example, 4√256 = 4 because 44 = 256 (a perfect 4th power). The US financial industry also uses 4th roots when calculating quarterly compound growth rates.
The most useful perfect squares to know: √1 = 1, √4 = 2, √9 = 3, √16 = 4, √25 = 5, √36 = 6, √49 = 7, √64 = 8, √81 = 9, √100 = 10, √121 = 11, √144 = 12, √169 = 13, √196 = 14, √225 = 15. Memorizing up to √225 covers most algebra, geometry, and standardized-test problems you will encounter. The full reference table earlier on this page extends this list to n = 25.