Root Calculator

Calculate Roots

Choose square, cube, or nth root and see the root together with the matching exponent relationship.

Even roots of negative numbers are complex; odd roots can be real.

Root result

Root value Enter a number Formula and power check will appear here.

What is a Root?

A root of a number is a value that, when multiplied by itself a certain number of times, gives the original number. The nth root of x is the number that, when raised to the power n, equals x.

The most common roots are the square root (2nd root) and cube root (3rd root). Square roots are used extensively in geometry, physics, and statistics, while cube roots appear in volume calculations and engineering.

How Root Calculations Work

Follow these detailed steps:

  1. Step 1: Identify the Radicand and Index
    In āæāˆša, a is the radicand (number under root) and n is the index (degree of root). āˆš16 is square root (n=2), Ā³āˆš8 is cube root.
  2. Step 2: Calculate the Root
    Find the number that, when raised to the power n, equals a. āˆš16 = 4 because 4Ā² = 16. Ā³āˆš8 = 2 because 2Ā³ = 8.
  3. Step 3: Simplify if Possible
    āˆš50 = āˆš(25Ɨ2) = 5āˆš2. Look for perfect squares (4, 9, 16, 25...) that divide evenly into the radicand.

Formulas

Square Root: āˆšx = x^(1/2)

Cube Root: āˆ›x = x^(1/3)

Nth Root: āæāˆšx = x^(1/n)

Relationship: (āæāˆšx)^n = x

Key Properties:

  • āˆš(a Ɨ b) = āˆša Ɨ āˆšb
  • āˆš(a/b) = āˆša / āˆšb
  • āˆšxĀ² = |x| (absolute value)
  • For even roots of negative numbers: use imaginary numbers

Examples

Square Root of 144

Problem: Calculate āˆš144

Solution:

  1. Find a number that multiplied by itself equals 144
  2. 12 Ɨ 12 = 144
  3. Result: āˆš144 = 12

Cube Root of 27

Problem: Calculate āˆ›27

Solution:

  1. Find a number that multiplied by itself twice equals 27
  2. 3 Ɨ 3 Ɨ 3 = 27
  3. Result: āˆ›27 = 3

Fourth Root of 81

Problem: Calculate ā“āˆš81

Solution:

  1. Find a number that raised to the 4th power equals 81
  2. 3 Ɨ 3 Ɨ 3 Ɨ 3 = 81
  3. Result: ā“āˆš81 = 3

Quick Calculation Tips

  • āˆš(aƗb) = āˆša Ɨ āˆšb - useful for simplification
  • āˆšaĀ² = |a| (absolute value), not just a
  • Odd roots of negative numbers are negative: Ā³āˆš(-8) = -2
  • Even roots of negative numbers are undefined in real numbers

Common Mistakes to Avoid

  • āˆš(a + b) ā‰  āˆša + āˆšb
    āˆš(9 + 16) = āˆš25 = 5, but āˆš9 + āˆš16 = 3 + 4 = 7. Roots don't distribute over addition.
  • Square root vs. principal root
    Equation xĀ² = 9 has TWO solutions: x = Ā±3. But āˆš9 = 3 (principal root only).

Frequently Asked Questions

What is the square root of a negative number?

The square root of a negative number is an imaginary number. For example, āˆš(-1) = i, where i is the imaginary unit. The square root of -4 is 2i. This calculator returns NaN for even roots of negative numbers.

What's the difference between āˆšxĀ² and (āˆšx)Ā²?

āˆšxĀ² = |x| (always positive, the absolute value of x), while (āˆšx)Ā² = x (only defined for x ā‰„ 0, equals x). The order of operations matters!

Can you take an odd root of a negative number?

Yes! Odd roots of negative numbers are defined and result in negative numbers. For example, āˆ›(-8) = -2 because (-2)Ā³ = -8.

How do I simplify square roots?

To simplify āˆšx, factor out perfect squares. For example, āˆš72 = āˆš(36 Ɨ 2) = 6āˆš2. Look for the largest perfect square that divides evenly into the number under the radical.

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