# Square root calculator: how to find square, cube and nth roots and simplify radicals

> A root undoes a power: the square root of 49 is 7 because 7 × 7 = 49. This guide shows how to estimate a root by hand, simplify √72 to 6√2, find cube and nth roots, spot perfect squares and avoid the mistake of splitting √(a + b).

Published 2026-11-07 by Toolcloak. 4 minute read. Original: https://toolcloak.com/articles/square-root-calculator-how-to-find-square-cube-and-nth-roots-and-simplify-radicals
Related free tool: [Root Calculator (Square, Cube & Nth Root)](https://toolcloak.com/tools/root-calculator)

What is the square root of 72? A calculator says 8.485281…, which is hardly useful in an exam or a drawing. In exact form it is 6√2, and once you see why, you can simplify any square root yourself.

> **In short**
> - The square root of x is the number that gives x when multiplied by itself. √49 = 7 because 7 × 7 = 49.
> - To simplify √72, pull out the largest perfect square factor: 72 = 36 × 2, so √72 = 6√2.
> - The nth root of x equals x raised to the power 1/n. The cube root of 27 is 3.

## What a root is

Roots are the reverse of powers. Squaring 7 gives 49, so the **square root** of 49 is 7. Cubing 3 gives 27, so the **cube root** of 27 is 3. In general, the **nth root** of x is the number that you multiply by itself n times to get x. It is written ⁿ√x, and it equals x^(1/n).

Try it: [Root Calculator (Square, Cube & Nth Root)](https://toolcloak.com/tools/root-calculator). Enter a number and the root degree (2 for a square root, 3 for a cube root). You see the result, the simplified radical form, and whether the number is a perfect power.

## Perfect squares you should know

A perfect square is a whole number whose square root is also a whole number. Knowing the first fifteen makes estimating and simplifying much faster.

| n | n² | n | n² | n | n² |
| --- | --- | --- | --- | --- | --- |
| 1 | 1 | 6 | 36 | 11 | 121 |
| 2 | 4 | 7 | 49 | 12 | 144 |
| 3 | 9 | 8 | 64 | 13 | 169 |
| 4 | 16 | 9 | 81 | 14 | 196 |
| 5 | 25 | 10 | 100 | 15 | 225 |

*The square root of any number in the n² column is the number beside it. √169 = 13.*

Perfect cubes follow the same idea: 1, 8, 27, 64, 125, 216, 343, 512, 729 and 1000 are the cubes of 1 to 10.

## How to estimate a square root by hand

Say you need √50. Find the two perfect squares around it: 49 (which is 7²) and 64 (which is 8²). So √50 sits between 7 and 8, and because 50 is only 1 above 49, it is very close to 7. A good estimate is 7.07, and the exact value is **7.0711** to four decimal places.

A refinement that works well: guess g, then average g with x ÷ g. For √50 start with 7. Then 50 ÷ 7 = 7.143 and the average of 7 and 7.143 is 7.071, already accurate to three decimals.

## How to simplify a square root

A square root is in simplest form when no perfect square (other than 1) divides the number inside. The method uses prime factors.

1. Break the number into prime factors. 72 = 2 × 2 × 2 × 3 × 3.
2. Group the factors in identical pairs. Here: (2 × 2), (3 × 3), and one 2 left over.
3. Move one number from each pair outside the root, and keep the leftovers inside. Outside: 2 × 3 = 6. Inside: 2.
4. Write the result: √72 = 6√2.

| Number | Largest square factor | Simplified | Decimal |
| --- | --- | --- | --- |
| √50 | 25 | 5√2 | 7.0711 |
| √72 | 36 | 6√2 | 8.4853 |
| √98 | 49 | 7√2 | 9.8995 |
| √200 | 100 | 10√2 | 14.1421 |
| √45 | 9 | 3√5 | 6.7082 |
| √17 | 1 | √17 | 4.1231 |

*√17 cannot be simplified because 17 is prime. The calculator shows this as √17.*

## Cube roots and higher roots

The same idea extends to any degree. The cube root of 64 is 4, because 4 × 4 × 4 = 64. The fourth root of 81 is 3, because 3 × 3 × 3 × 3 = 81. For numbers that are not perfect powers you get a decimal, such as the cube root of 2, which is about 1.2599.

**Negative numbers.** An odd root of a negative number is fine: the cube root of −8 is −2, because (−2) × (−2) × (−2) = −8. An even root of a negative number has no real answer, since no real number multiplied by itself an even number of times is negative. That is why √−4 shows an error in a real-number calculator.

## Rules that save time

- **Product rule:** √(a × b) = √a × √b. This is what makes √72 = √36 × √2 = 6√2 work.
- **Quotient rule:** √(a ÷ b) = √a ÷ √b. So √(9 ÷ 4) = 3 ÷ 2 = 1.5.
- **Power link:** the nth root of x equals x^(1/n), so √x = x^0.5.
- **A warning:** √(a + b) is **not** √a + √b. √9 + √16 = 3 + 4 = 7, but √(9 + 16) = √25 = 5.

## Where roots show up

The diagonal of a square with side 10 cm is 10√2, about 14.14 cm. The side of a square with area 150 m² is √150, about 12.25 m. The Pythagorean theorem for a 5 and 12 right triangle gives a hypotenuse of √(25 + 144) = √169 = 13. Standard deviation in statistics is a square root, and so are many physics formulas for speed and frequency.

## Frequently asked questions

### How do I find the square root of a number?

Find the number that multiplied by itself gives your number. √49 = 7 because 7 × 7 = 49. For non-perfect squares, estimate between two perfect squares or use a calculator.

### How do I simplify √72?

Factor 72 as 36 × 2, take the square root of 36 (which is 6) outside the radical, and you get 6√2.

### What is the cube root of 27?

3, because 3 × 3 × 3 = 27.

### Can you take the square root of a negative number?

Not in the real numbers. No real number squared is negative. The result is an imaginary number such as √−4 = 2i, which a real-number calculator does not show.

### What is the difference between a root and a power?

They are inverses. Squaring 7 gives 49, and the square root of 49 gives 7 back. The nth root of x is the same as x^(1/n).

### Is √(a + b) equal to √a + √b?

No. √9 + √16 = 7 but √(9 + 16) = √25 = 5. Roots do not split over addition, only over multiplication and division.