Root Calculator: Find Square Root, Cube Root & Nth Root

Finding the root of a number, whether it’s a simple square root or a more complex nth root, is a fundamental math operation. Our calculator provides a quick and accurate way to find any root you need, instantly.

Modern Root Calculator

Calculate the square root, cube root, or any other integer root of a number.

How to Use Our Root Calculator

Our tool is designed to be straightforward. Here’s a simple guide to the input fields:

Degree (n): This is the type of root you want to find. For a square root, enter 2. For a cube root, enter 3. For any other root (like the 4th root or 5th root), enter the desired number here.
Number (x): This is the number you want to find the root of. This number is called the ‘radicand’.
Calculate: After entering the degree and the number, click this button to get your answer.

Understanding Your Results

The calculator solves the equation $s q r t [n] x = y$ . The result, ‘y’, is the number that, when multiplied by itself ‘n’ times, equals your original number ‘x’.

Finding a root is the inverse operation of raising a number to a power (exponents). The relationship can be expressed like this:

If $s q r t [n] x = y$ , then $y^{n} = x$ .

Let’s break this down with a few examples:

Square Root (n=2): If you calculate the square root of 9 ( $s q r t [2] 9$ ), the result is 3. This is because $3^{2}$ (or $3 t im es 3$ ) equals 9.
Cube Root (n=3): If you calculate the cube root of 27 ( $s q r t [3] 27$ ), the result is 3. This is because $3^{3}$ (or $3 t im es 3 t im es 3$ ) equals 27.
4th Root (n=4): If you calculate the 4th root of 16 ( $s q r t [4] 16$ ), the result is 2. This is because $2^{4}$ (or $2 t im es 2 t im es 2 t im es 2$ ) equals 16.

Our calculator finds the principal root, which is the positive real number that solves the equation.

Properties of Roots (Radicals)

Understanding the rules of how roots (also called radicals) work can help you simplify problems manually.

Property Name	Formula	Explanation & Example
Product Rule	$s q r t [n] a c d o t b = s q r t [n] a c d o t s q r t [n] b$	The root of a product is the product of the roots. This helps simplify radicals. Example: $s q r t 36 = s q r t 4 c d o t 9 = s q r t 4 c d o t s q r t 9 = 2 c d o t 3 = 6$ .
Quotient Rule	$s q r t [n] f r a c a b = f r a c s q r t [n] a s q r t [n] b$	The root of a quotient (fraction) is the quotient of the roots. Example: $s q r t [3] f r a c 648 = f r a c s q r t [3] 64 s q r t [3] 8 = f r a c 42 = 2$ .
Fractional Exponent Rule	$s q r t [n] x^{m} = x^{f r a c m n}$	A root can be expressed as a fractional exponent. The degree of the root becomes the denominator of the fraction. Example: $s q r t [3] x^{2} = x^{f r a c 23}$ .
Nested Roots Rule	$s q r t [m] s q r t [n] x = s q r t [m c d o t n] x$	To simplify a root of a root, you can multiply the degrees of the roots together. Example: $s q r t [3] s q r t 64 = s q r t [3 c d o t 2] 64 = s q r t [6] 64 = 2$ .

Frequently Asked Questions

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

The difference is the degree of the root, which tells you how many times the resulting number must be multiplied by itself to equal the original number.

Square Root ( $s q r t x$ ): The degree is 2. It asks, “What number times itself equals x?” It’s used to find the side length of a square given its area. By convention, the ‘2’ is usually omitted from the radical symbol.
Cube Root ( $s q r t [3] x$ ): The degree is 3. It asks, “What number times itself three times equals x?” It’s used to find the side length of a cube given its volume.

Feature	Square Root	Cube Root
Notation	$s q r t x$ or $s q r t [2] x$	$s q r t [3] x$
Inverse Operation	Squaring ( $y^{2}$ )	Cubing ( $y^{3}$ )
Example	$s q r t 25 = 5$ (because $5^{2} = 25$ )	$s q r t [3] 125 = 5$ (because $5^{3} = 125$ )
Common Use	Geometry (Area), Pythagorean theorem	Geometry (Volume)

Can you take the root of a negative number?

It depends on the degree (n) of the root.

Odd Degree (n=3, 5, 7…): Yes. You can take an odd root of a negative number, and the result will also be negative.
- Concrete Example: $s q r t [3] -8 = - 2$ , because $(- 2) t im es (- 2) t im es (- 2) = - 8$ .
Even Degree (n=2, 4, 6…): No, not in the real number system. There is no real number that, when multiplied by itself an even number of times, results in a negative number. (e.g., $2 t im es 2 = 4$ and $(- 2) t im es (- 2) = 4$ ). The answer involves imaginary numbers (e.g., $s q r t -1 = i$ ).

What is a “principal root”?

For any positive number, there are technically two square roots: one positive and one negative. For example, both $5^{2}$ and $(- 5)^{2}$ equal 25. The principal root is the positive real number solution. When people refer to “the square root of 25,” they are almost always referring to the principal root, which is 5. Our calculator provides the principal root.

When are roots used in real life?

Roots are essential in many fields beyond math class:

Construction & Engineering: The Pythagorean theorem ( $a^{2} + b^{2} = c^{2}$ ) uses square roots to calculate lengths and distances, like the diagonal of a room or the required length of a support beam.
Finance: Calculating the average rate of return on an investment over several years uses geometric means, which involves nth roots.
Science: Scientists use roots in formulas for calculating things like the speed of waves, orbital periods of planets, and measurements in quantum physics.
Statistics: The standard deviation, a key measure of data spread, is calculated using a square root.

How do I write a root as an exponent?

Any root can be rewritten as a fractional exponent. This is one of the most useful properties for simplifying complex expressions. The rule is:

$s q r t [n] x = x^{f r a c 1 n}$

Concrete Examples:

$s q r t 5 = 5^{f r a c 12}$
$s q r t [3] 10 = 1 0^{f r a c 13}$
$s q r t [8] y = y^{f r a c 18}$

What is the root of 1?

The nth root of 1 is always 1, regardless of the degree (n). This is because 1 multiplied by itself any number of times is still 1. ( $1^{n} = 1$ ).

What is the root of 0?

The nth root of 0 is always 0, regardless of the degree (n). This is because 0 multiplied by itself any number of times is still 0. ( $0^{n} = 0$ ).

How can I estimate a square root without a calculator?

You can estimate a square root by finding the two perfect squares the number lies between.

Concrete Example: Estimate $s q r t 55$ .

Find the closest perfect squares. We know that $7^{2} = 49$ and $8^{2} = 64$ .
Locate the number. 55 is between 49 and 64. Therefore, $s q r t 55$ must be between 7 and 8.
Estimate the decimal. Since 55 is a bit closer to 49 than it is to 64, the answer will be closer to 7. A good estimate would be around 7.4.
Check the estimate: $7.4 t im es 7.4 = 54.76$ . This is very close to 55, making it a great estimation.

Why can’t the degree of a root be 0?

If the degree of a root (n) were 0, it would correspond to a fractional exponent with 0 in the denominator ( $s q r t [0] x = x^{f r a c 10}$ ). Division by zero is undefined in mathematics, so a 0th root is not a defined operation.

Is a radical the same as a root?

Yes, the terms are often used interchangeably. A radical is the symbol used to denote a root ( $s q r t p han t o m x$ ). The entire expression, such as $s q r t 25$ , is also called a radical expression. “Finding the root” and “simplifying the radical” mean the same thing.

Now that you’ve calculated a root, you might want to perform the inverse operation with our Exponent Calculator. If you’re working on geometry problems, our Pythagorean Theorem Calculator can help you find the lengths of a right triangle’s sides.

Creator

Ismael Vargas

An experienced software developer specializing in React, JavaScript, Django and Python, with more than six years’ expertise building full‑stack applications, data visualizations and cloud‑hosted solutions. He has a strong background in API integration, testing, and AWS services, delivering polished web products.

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