What is a Root?

In mathematics, finding the root of a number is the inverse operation of raising a number to a power. While "square root" is the most famous, you can find any "nth" root of a number. The nth root of a number 'x' is a number 'y' such that yⁿ = x.

• The Square Root (2nd root) of 25 is 5, because 5² (5 × 5) = 25.
• The Cube Root (3rd root) of 27 is 3, because 3³ (3 × 3 × 3) = 27.
• The 4th Root of 16 is 2, because 2⁴ (2 × 2 × 2 × 2) = 16.

This concept is a cornerstone of algebra and has wide-ranging applications in various fields of science and engineering.

Perfect Roots and Irrational Numbers

• Perfect Roots: A number is a "perfect nth root" if its nth root is a whole number (an integer). For example, 81 is a perfect 4th root because its 4th root is 3. Our calculator will immediately tell you if the number you have entered is a perfect root for the selected degree.
• Irrational Numbers: When you take a root of a number that is not a perfect root, the result is often an irrational number. An irrational number is a number that cannot be expressed as a simple fraction—it has an infinite, non-repeating decimal expansion. For example, the cube root of 10 is approximately 2.15443469...

How to Calculate a Root

Calculating higher-order roots manually is complex, but the underlying mathematical principle involves fractional exponents.

The nth root of x = x^(1/n)

This means finding the cube root of a number is the same as raising that number to the power of 1/3. This is the principle that scientific calculators and our online tool use to find roots instantly and accurately.

Real-World Applications of Roots

Roots are not just an abstract concept; they are fundamental to many fields.

• Geometry and Engineering (Square & Cube Roots): The Pythagorean Theorem (a² + b² = c²) uses square roots to find distances. Cube roots are essential for calculations involving the volume of three-dimensional objects. For example, if you know the volume of a cube, you can find the length of its sides by taking the cube root.
• Finance (nth Roots): Higher-order roots are crucial in finance for calculating the average rate of return over multiple years. The Compound Annual Growth Rate (CAGR), a key investment metric, uses the nth root, where 'n' is the number of years.
• Physics and Science: Many physical laws involve roots. For example, Kepler's third law of planetary motion uses a 3/2 power (which involves a square root) to relate a planet's orbital period to its distance from the sun.

Frequently Asked Questions (FAQs)

1. How do I use the Root Calculator?

First, select the root you want to find from the dropdown menu (e.g., Square Root, Cube Root, etc.). Then, enter the number you want to find the root of into the input box. The calculator will instantly display the result.

2. What is the root of a negative number?

This depends on the root index. You can take an *odd* root (3rd, 5th, etc.) of a negative number. For example, the cube root of -27 is -3, because (-3) × (-3) × (-3) = -27. However, you cannot take an *even* root (2nd, 4th, etc.) of a negative number within the real number system. Our calculator will show an "Invalid" error in such cases.

3. Why does a positive number have two square roots but only one cube root?

A positive number has two *even* roots (one positive, one negative) because a negative number raised to an even power becomes positive (e.g., (-5)² = 25). However, a negative number raised to an *odd* power remains negative (e.g., (-3)³ = -27). Therefore, a positive number only has one positive odd root.