Square Root Calculator

About this tool

The Square Root Calculator shows the square root, cube root and an nth root of your choice (n from 2 to 100) side by side. Each value is exact when the root ends and fits in 40 characters, as in √2.25 = 1.5 and ∛1000 = 10; otherwise it is rounded to 10 significant digits, as in √72 ≈ 8.485281374. For whole numbers up to 1,000,000,000,000,000 it also gives the simplified radical form by pulling out the largest perfect power: √72 = √(36 × 2) = 6√2 and ∛54 = ∛(27 × 2) = 3∛2. It tells you whether a whole number is a perfect square or cube, and between which squares and cubes it lies, for example 8² = 64 and 9² = 81 around 72. Odd roots of negative numbers are real, so ∛(−27) = −3. Even roots of negative numbers are not real; for the square root the calculator shows the imaginary result instead, such as √(−16) = 4i. Decimal results come from the browser's floating-point math, while exactness checks use whole-number arithmetic. It all runs in your browser.

Why use it

• Square, cube and nth roots in one view, so you do not need a separate cube root calculator.

• Simplifies radicals and shows the perfect-power factor that comes out.

• Marks every result as exact or rounded, and returns exact roots of big perfect squares, such as the 20-digit square root of a 39-digit number.

• Handles negative numbers: real odd roots and imaginary square roots.

• Shows the nearest perfect squares and cubes, which helps you estimate roots by hand.

How to use

1. Type a number: a whole number, a decimal such as 2.25, a negative number or e-notation such as 1e6.
2. Optionally change n for the third card. It starts at 4 and accepts whole numbers from 2 to 100.
3. Read the square root, cube root and nth root cards; each says whether the value is exact or rounded to 10 significant digits.
4. For whole numbers, read the simplified radical form, such as 6√2.
5. Check whether the number is a perfect square or cube and which ones it lies between.
6. Press Copy results to copy all values and lines.

When it helps

• Simplifying radicals in algebra homework, such as √50 = 5√2.
• Finding the side of a square from its area: a square room of 144 sq ft has 12 ft sides.
• Finding the edge of a cube from its volume: a 1,000 cm³ cube has 10 cm edges.
• Checking whether a number is a perfect square or perfect cube.
• Taking nth roots for growth rates, such as the yearly rate that doubles money in 5 years.

Examples

Tips

• Estimate a square root from the perfect squares around it: 72 lies between 64 and 81, so √72 is between 8 and 9, closer to 8.
• A radical is fully simplified when no perfect-square factor (for √) or perfect-cube factor (for ∛) other than 1 is left under the sign.
• The nth root is the same as the power 1/n, so the 4th root of 16 is 16^(1/4) = 2.
• Average yearly growth over n years is the nth root of the total growth factor minus 1: doubling in 5 years gives ⁵√2 ≈ 1.148698355, about 14.87% a year.
• Every positive number has two square roots, 12 and −12 for 144; the √ sign and this calculator give the positive one.

Frequently Asked Questions

Glossary

Radicand

The number under the root sign; in √72 the radicand is 72.

Index

The small number that says which root to take; the cube root ∛ has index 3.

Perfect square

A whole number equal to a whole number squared, such as 49 = 7².

Simplified radical form

A root with no perfect-power factor left under the sign, such as 6√2 for √72.

Principal root

The non-negative root that the √ sign stands for.

Imaginary unit

The number i, defined by i² = −1, used to write square roots of negative numbers.