Nth Root Calculator

Easily find any root of a number, from square roots to complex nth roots.

Enter the root degree (n) and the number (radicand) to compute the result. This tool is perfect for mathematical calculations in academics and professional fields.

Enter the root's degree (n > 0)

Enter the number to find the root of

Examples

Click on an example to load it into the calculator.

Square Root of a Perfect Square

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Find the square root (n=2) of 81.

Degree (n): 2

Number: 81

Cube Root of a Negative Number

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Find the cube root (n=3) of -64.

Degree (n): 3

Number: -64

4th Root of a Number

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Find the 4th root (n=4) of 16.

Degree (n): 4

Number: 16

5th Root of a Larger Number

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Find the 5th root (n=5) of 243.

Degree (n): 5

Number: 243

Other Titles
Understanding the Root Calculator: A Comprehensive Guide
Master the concept of roots, from basic square roots to complex nth roots, and learn how to apply them in various mathematical and real-world scenarios.

What is a Root? Mathematical Foundation and Concepts

  • A root is the inverse operation of exponentiation
  • It helps find the base number that, when raised to a power, equals a given value
  • Roots are fundamental in algebra, geometry, and calculus
In mathematics, finding a root is the process of reversing exponentiation. When we ask for the 'nth root' of a number 'x', we are looking for a number 'y' that, when multiplied by itself 'n' times, gives us 'x'. This relationship is formally written as yⁿ = x.
Key Terminology
The expression ⁿ√x consists of three parts: Radical Symbol (√), the symbol used to denote a root operation; Radicand (x), the number under the radical symbol for which the root is being calculated; and Degree (n), also known as the index, which indicates which root to take. If no degree is specified, it is implicitly a square root (n=2).
For example, in the expression ³√27, the degree is 3, and the radicand is 27. We are looking for a number that, when cubed, equals 27. The answer is 3, because 3³ = 27.

Fundamental Root Examples

  • Square Root (n=2): √64 = 8, because 8² = 64.
  • Cube Root (n=3): ³√-8 = -2, because (-2)³ = -8.
  • 4th Root (n=4): ⁴√81 = 3, because 3⁴ = 81.

Step-by-Step Guide to Using the Nth Root Calculator

  • Enter the degree and the number correctly
  • Understand the input fields and their requirements
  • Interpret the results for both positive and negative numbers
Our Nth Root Calculator simplifies finding roots to just a few clicks. Follow these steps for accurate calculations.
Input Instructions
Step 1: Enter the Degree (n): In the 'Degree (n)' field, input the type of root you want to find. For a square root, enter 2. For a cube root, enter 3, and so on. The degree must be a positive number.
Step 2: Enter the Number (Radicand): In the 'Number (Radicand)' field, input the number you wish to find the root of. This can be a positive or negative number.
Step 3: Calculate: Click the 'Calculate Root' button to process the inputs.
Step 4: Review the Result: The calculated root will appear in the 'Result' section. The tool automatically handles the complexities of negative radicands and odd/even degrees.

Practical Usage Examples

  • To find the square root of 144, enter Degree=2 and Number=144. Result: 12.
  • To find the cube root of -125, enter Degree=3 and Number=-125. Result: -5.
  • For the 10th root of 1024, enter Degree=10 and Number=1024. Result: 2.

Real-World Applications of Root Calculations

  • Roots are essential in finance for calculating interest rates
  • They are used in physics to model phenomena like oscillations and waves
  • In geometry, roots help determine lengths and scale dimensions
Root calculations are not just abstract mathematical exercises; they have significant applications in various professional and scientific fields.
Finance and Economics
To determine the average annual rate of return on an investment that has grown over several years, one must calculate the nth root of the total growth ratio. This is crucial for analyzing investment performance.
Engineering and Physics
In physics, the period of a pendulum is proportional to the square root of its length. In electrical engineering, the root mean square (RMS) value of an AC voltage involves a square root. These calculations are fundamental to designing and analyzing physical systems.
Geometry and Architecture
The Pythagorean theorem, c = √(a² + b²), uses a square root to find the length of a hypotenuse. When scaling objects, roots are used to find the correct scaling factor. For instance, to double the volume of a sphere, you must increase its radius by a factor of ³√2.

Industry-Specific Examples

  • An investment triples in 5 years. The average annual growth rate is ⁵√3 - 1 ≈ 24.57%.
  • A civil engineer needs to find the side length of a square plot of land with an area of 20,000 sq ft. The length is √20000 ≈ 141.42 ft.

Common Misconceptions and Correct Methods

  • Clarifying the meaning of the principal root
  • Understanding why roots do not distribute over addition
  • Handling even and odd roots of negative numbers correctly
Misconception: The square root of a number is always positive and negative.
This is a common point of confusion. The equation x² = 9 has two solutions, x = 3 and x = -3. However, the radical symbol (√) specifically denotes the principal root, which is the non-negative one. Therefore, √9 is defined strictly as 3. To represent both solutions, one must use the notation ±√9.
Misconception: You can distribute a root over addition or subtraction.
A frequent algebraic mistake is assuming that √(a + b) = √a + √b. This is incorrect. For example, √(9 + 16) = √25 = 5. However, √9 + √16 = 3 + 4 = 7. The two results are not equal.
Misconception: It's impossible to find the root of a negative number.
This is only partially true. You can find an odd-indexed root of a negative number (e.g., ³√-27 = -3). However, an even-indexed root (like a square root or 4th root) of a negative number does not yield a real number; the result is a complex number.

Key Clarifications

  • √16 = 4 (the principal root), not ±4.
  • √(a² + b²) ≠ a + b.
  • ⁵√-32 = -2 is a valid real number.

Mathematical Derivation and Fractional Exponents

  • Expressing roots as fractional exponents for easier manipulation
  • Applying exponent rules to simplify complex root expressions
  • Understanding the mathematical engine behind the calculator
The calculation of roots is deeply connected to the rules of exponents. Any root can be expressed as a fractional exponent, which simplifies many algebraic operations.
The Fractional Exponent Identity
ⁿ√x = x^(1/n)
This identity is incredibly powerful. It means that finding the nth root of a number is the same as raising that number to the power of 1/n. Our calculator uses this principle for its computations, often leveraging the Math.pow() function or its equivalent in programming languages.
For example, to find the 4th root of 625, we can compute 625^(1/4). Since 5⁴ = 625, we know that 625^(1/4) = 5. This method is numerically stable and efficient for computers to handle.

Calculation Example

  • Calculate the 6th root of 729.
  • Using fractional exponents: 729^(1/6).
  • We can recognize that 729 = 3⁶.
  • So, (3⁶)^(1/6) = 3^(6 * 1/6) = 3¹ = 3.
  • The calculator confirms this by computing Math.pow(729, 1/6), which returns 3.