Cube and Cube Root Calculator

About Cube and Cube Root Calculator

Welcome to the Cube and Cube Root Calculator, a powerful online tool that calculates the cube (n³) and cube root (√³n) of any number instantly. Whether you need to find the volume of a cube, solve polynomial equations, or verify perfect cubes, this calculator provides step-by-step solutions with interactive 3D visualization.

What is a Cube of a Number?

The cube of a number n is the result of multiplying the number by itself three times: n × n × n = n³. Geometrically, this represents the volume of a cube with side length n units.

For example:

• 2³ = 2 × 2 × 2 = 8
• 5³ = 5 × 5 × 5 = 125
• 10³ = 10 × 10 × 10 = 1,000
• (-3)³ = (-3) × (-3) × (-3) = -27

What is a Cube Root?

The cube root of a number x is a value that, when multiplied by itself three times, equals x. It is written as √³x or x^1/3. The cube root is the inverse operation of cubing a number.

For example:

• √³8 = 2 (because 2³ = 8)
• √³27 = 3 (because 3³ = 27)
• √³125 = 5 (because 5³ = 125)
• √³-64 = -4 (because (-4)³ = -64)

What is a Perfect Cube?

A perfect cube is an integer that can be expressed as the cube of another integer. Perfect cubes have exact integer cube roots, making them particularly useful in mathematical calculations.

List of Perfect Cubes (1-20)

The first 20 perfect cubes are:

• 1³ = 1
• 2³ = 8
• 3³ = 27
• 4³ = 64
• 5³ = 125
• 6³ = 216
• 7³ = 343
• 8³ = 512
• 9³ = 729
• 10³ = 1,000
• 11³ = 1,331
• 12³ = 1,728
• 13³ = 2,197
• 14³ = 2,744
• 15³ = 3,375
• 16³ = 4,096
• 17³ = 4,913
• 18³ = 5,832
• 19³ = 6,859
• 20³ = 8,000

How to Calculate Cube and Cube Root

How to Calculate Cube

1. Enter your number: Input any integer, decimal, or negative number into the calculator.
2. Select calculation mode: Choose "Cube Only (n³)" or "Both Cube and Cube Root" to calculate the cube.
3. Click Calculate: Press the Calculate button to get instant results.
4. Review results: View the cube value with step-by-step calculation breakdown and 3D visualization.

How to Calculate Cube Root

1. Enter your number: Input the number for which you want to find the cube root.
2. Select calculation mode: Choose "Cube Root Only (√³n)" or "Both Cube and Cube Root".
3. Click Calculate: Press the Calculate button to compute the cube root.
4. Review results: See the cube root value with step-by-step solution and perfect cube detection.

Cube Root of Negative Numbers

Unlike square roots, cube roots are defined for negative numbers. The cube root of a negative number is negative because a negative number multiplied by itself three times yields a negative result.

For example:

• √³(-8) = -2 because (-2) × (-2) × (-2) = -8
• √³(-27) = -3 because (-3) × (-3) × (-3) = -27
• √³(-1000) = -10 because (-10) × (-10) × (-10) = -1000

Properties of Cubes and Cube Roots

Properties of Cubes

• The cube of a positive number is positive
• The cube of a negative number is negative
• The cube of zero is zero
• For any real number n: (n³)^1/3 = n
• Product rule: (a × b)³ = a³ × b³
• Quotient rule: (a / b)³ = a³ / b³

Properties of Cube Roots

• Every real number has exactly one real cube root
• The cube root of a positive number is positive
• The cube root of a negative number is negative
• √³(a × b) = √³a × √³b
• √³(a / b) = √³a / √³b

Applications of Cubes and Cube Roots

Geometry and Volume

Cubes are essential for calculating the volume of cube-shaped objects:

• Volume of a cube = side³
• Finding side length from volume = √³volume

Physics and Engineering

• Calculating the volume of containers and storage spaces
• Determining displacement in fluid mechanics
• Computing density relationships

Mathematics and Algebra

• Solving cubic equations
• Simplifying radical expressions
• Working with polynomial functions

Frequently Asked Questions

What is a cube of a number?

The cube of a number n is the result of multiplying the number by itself three times: n × n × n = n³. For example, the cube of 3 is 3³ = 3 × 3 × 3 = 27. Cubing a number gives the volume of a cube with side length n.

What is a cube root?

The cube root of a number x is a value that, when multiplied by itself three times, equals x. Written as √³x or x^1/3, the cube root of 27 is 3 because 3 × 3 × 3 = 27. Unlike square roots, cube roots exist for negative numbers too.

What is a perfect cube?

A perfect cube is an integer that can be expressed as the cube of another integer. Examples include 1 (1³), 8 (2³), 27 (3³), 64 (4³), and 125 (5³). Perfect cubes have exact integer cube roots, making them easy to work with in calculations.

How do you calculate cube root of negative numbers?

For negative numbers, the cube root follows the rule: √³(-x) = -√³x. This is because a negative number cubed produces a negative result. For example, √³(-27) = -3 because (-3) × (-3) × (-3) = -27. This is different from square roots, which are not defined for negative real numbers.

What is the formula for cube and cube root?

The cube formula is n³ = n × n × n. The cube root formula is √³x = x^1/3. These are inverse operations: if n³ = x, then √³x = n. For example, 5³ = 125 and √³125 = 5. This calculator computes both instantly with step-by-step solutions.

Additional Resources

• Cube (Algebra) - Wikipedia
• Cube Root - Wikipedia

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