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How to Cube A Number Without Calculator

Reviewed by Calculator Editorial Team

Cubing a number means multiplying the number by itself three times. While calculators make this operation quick and easy, there are several methods you can use to cube a number without one. This guide explains these methods, provides examples, and includes a free online calculator for verification.

What is cubing a number?

Cubing a number is a mathematical operation that involves multiplying the number by itself three times. The result is called a cube. For example, 3 cubed (written as 3³) is calculated as 3 × 3 × 3 = 27.

Formula: a³ = a × a × a

The cube of a number is different from squaring it (which is multiplying the number by itself twice). Cubing is used in various mathematical fields, including algebra, geometry, and calculus.

Methods to cube a number without calculator

There are several methods you can use to cube a number without a calculator. Here are the most common ones:

Method 1: Repeated Multiplication

The simplest method is to multiply the number by itself three times. For example, to find 4³:

  1. Multiply 4 by 4: 4 × 4 = 16
  2. Multiply the result by 4 again: 16 × 4 = 64

So, 4³ = 64.

Method 2: Using the Formula for Cubes of Binomials

For numbers that can be expressed as (a + b), you can use the binomial expansion formula:

(a + b)³ = a³ + 3a²b + 3ab² + b³

For example, to find (2 + 3)³:

  1. Calculate a³: 2³ = 8
  2. Calculate 3a²b: 3 × (2²) × 3 = 3 × 4 × 3 = 36
  3. Calculate 3ab²: 3 × 2 × (3²) = 3 × 2 × 9 = 54
  4. Calculate b³: 3³ = 27
  5. Add all results: 8 + 36 + 54 + 27 = 125

So, (2 + 3)³ = 125, which matches 5³.

Method 3: Using the Difference of Cubes Formula

For numbers that can be expressed as (a - b), you can use the difference of cubes formula:

(a - b)³ = a³ - 3a²b + 3ab² - b³

For example, to find (5 - 2)³:

  1. Calculate a³: 5³ = 125
  2. Calculate -3a²b: -3 × (5²) × 2 = -3 × 25 × 2 = -150
  3. Calculate 3ab²: 3 × 5 × (2²) = 3 × 5 × 4 = 60
  4. Calculate -b³: -2³ = -8
  5. Add all results: 125 - 150 + 60 - 8 = 57

So, (5 - 2)³ = 57, which matches 3³.

Method 4: Using the Sum of Cubes Formula

For numbers that can be expressed as (a + b), you can use the sum of cubes formula:

a³ + b³ = (a + b)(a² - ab + b²)

For example, to find 3³ + 4³:

  1. Calculate a + b: 3 + 4 = 7
  2. Calculate a² - ab + b²: 9 - 12 + 16 = 13
  3. Multiply results: 7 × 13 = 91

So, 3³ + 4³ = 91.

Worked Examples

Let's look at some examples of cubing numbers without a calculator.

Example 1: Cubing a Single-Digit Number

Find 6³ using repeated multiplication:

  1. 6 × 6 = 36
  2. 36 × 6 = 216

So, 6³ = 216.

Example 2: Cubing a Two-Digit Number

Find 12³ using the binomial expansion formula:

  1. Express 12 as (10 + 2)
  2. Calculate (10 + 2)³ = 10³ + 3 × 10² × 2 + 3 × 10 × 2² + 2³
  3. Calculate each term: 1000 + 600 + 120 + 8 = 1728

So, 12³ = 1728.

Example 3: Cubing a Negative Number

Find (-3)³ using repeated multiplication:

  1. -3 × -3 = 9
  2. 9 × -3 = -27

So, (-3)³ = -27.

Frequently Asked Questions

What is the difference between squaring and cubing a number?

Squaring a number means multiplying it by itself once (a² = a × a), while cubing means multiplying it by itself twice (a³ = a × a × a). The cube of a number is always larger than its square.

Can I cube a negative number?

Yes, you can cube a negative number. The result will be negative if the original number is negative. For example, (-2)³ = -8.

What is the cube of zero?

The cube of zero is zero (0³ = 0). This is because any number multiplied by zero is zero.

Are there any shortcuts for cubing numbers?

Yes, there are several shortcuts such as using the binomial expansion formula, difference of cubes formula, and sum of cubes formula, which can simplify the calculation process.