Cube Root And Cube: Definition, Symbol, Methods to Calculate, and Solved Examples

Cube Root And Cube: Definition, Symbol, Methods to Calculate, and Solved Examples

A Cube Root And Cube of a number is the value that multiplies three times to equal the original. Learn methods to calculate the cube root of a number here.

Cube root: Suppose you are building a cube-shaped box and know its volume is 27 cubic units. How would you find the length of each side? The solution lies in the cube root.

A cube root works in reverse of cubing a number. For example, 3 × 3 × 3 = 27, so the cube root of 27 is 3. This simple yet powerful concept is used in real life to calculate dimensions, volumes, and much more. Let’s explore the concept of cube roots in detail here.

What is a Cube?

Before discussing what Cube Root And Cube are, let’s first understand what a cube is in math.

A cube is the result of multiplying a number by itself three times. In other words, when a number is raised to the power of 3, it is said to be “cubed.”

Example:

The cube of 2 is calculated as:

2× 2 × 2 = 8

So, 8 is the cube of 2.

Similarly, the cube of 4 is:

4× 4× 4=64

Hence, 64 is the cube of 4.

In mathematical notation, the cube of a number n is written as n³. The small 3 indicates that the number is multiplied by itself three times.

Unit Digits in Cube Numbers

The unit digit of a number’s cube follows specific patterns based on whether the number is odd or even.

Odd Numbers:

If a number is odd, its cube will also have an odd unit digit.

For example:

The cube of 3 is 27, and the unit digit of 27 is 7 (odd).

The cube of 5 is 125, and the unit digit of 125 is 5 (odd).

Even Numbers:

If a number is even, its cube will also have an even unit digit.

For example:

The cube of 2 is 8, and the unit digit of 8 is 8 (even).

The cube of 6 is 216, and the unit digit of 216 is 6 (even).

Now that we know what a cube is, let’s explore cube roots!

Cube Root Definition

A cube root of a number ‘x’ is a value ‘y’ such that when multiplied by itself three times (i.e., y × y × y), it equals x.

Mathematically, this is expressed as:

y³ = x, where x is the cube of y.

In other words, the cube root of a number is the value that, when cubed, gives the original number

For instance, the cube root of 512 is 8 because 8 × 8 × 8 = 512

Cube Root Symbol

The Cube Root And Cube is represented by the symbol ‘3√’​. Unlike the square root, which uses the symbol ‘√’, the cube root includes a small 3 to indicate that it represents the root of a number that has been cubed.

For example:

• The cube root of 5 is written as 3√5
• The cube root of 11 is written as 3√11

This notation helps differentiate cube roots from square roots.

Cube Root Formula

The cube root formula helps to find the cube root of any given number. If the cube of a number m is n, it can be expressed as:

m³ = n

To calculate the cube root of n, the formula is:

n^1/3 = m

This formula establishes the relationship between a number, its cube, and its cube root, making it useful in various mathematical calculations.

Cube Root And Cube

Number (a)	Cube (a³)	Cube Root (∛a)
1	1	1.000
2	8	1.260
3	27	1.442
4	64	1.587
5	125	1.710
6	216	1.817
7	343	1.913
8	512	2.000
9	729	2.080
10	1000	2.154
11	1331	2.224
12	1728	2.289
13	2197	2.351
14	2744	2.410
15	3375	2.466
16	4096	2.520
17	4913	2.571
18	5832	2.621
19	6859	2.668
20	8000	2.714

How to Find the Cube Root of a Number?

Finding the cube root of a number can be done using two main methods:

1. Prime Factorization Method
2. Estimation Method

Here’s an explanation with step-by-step details and new examples:

Prime Factorization Method

This method involves breaking the number into its prime factors. A number is a perfect cube if all the prime factors can be grouped into sets of three. The Cube Root And Cube is found by taking one factor from each group.

Steps to find cube root using prime factorization:

Step 1: Perform the prime factorization of the number.

Step 2: Group the prime factors into sets of three.

Step 3: Multiply one factor from each group to find the cube root.

Example 1: Find the cube root of 729.

Prime factorize 729: 729 = 3 × 3 × 3 × 3 × 3 ×3

Group the factors into sets of three: 729 = (3×3×3) × (3×3×3)

Take one factor from each group and multiply: 3×3=9

The cube root of 729 is 9.

Example 2: Find the cube root of 3375.

Prime factorize 3375: 3375=5 × 5 × 5 × 3 × 3 × 3

Group the factors into sets of three: 3375= (5×5×5) × (3×3×3)

Take one factor from each group and multiply: 5×3=15

The cube root of 3375 is 15.

Cube Root And Cube

Estimation Method

The estimation method is used to find the Cube Root And Cube of larger numbers by grouping digits and using properties of cubes.

Steps to find Cube Root And Cube using the estimation method

Step 1: Separate the number into groups of three digits, starting from the rightmost digit.

Step 2: Use the last digit of the number to determine the unit digit of the cube root (based on known cube values).

Step 3: Identify the range of cubes in which the remaining group of digits falls.

Step 4: Combine the results to find the cube root.

Tip: The following table can be helpful in determining the unit digit of a cube root

Last Digit of Number	0	1	2	3	4	5	6	7	8	9
Unit digit of Cube Root	0	1	8	7	4	5	6	3	2	9

Example 1: Find the Cube Root And Cube of 9261.

Group the digits: 9261=(9) (261)

Use the last digit (1) of the number 261:

• From the cube table, the cube root of a number ending in 1 also ends in 1.
• The unit digit of the cube root is 1.

Determine the range of cubes for the first group (9):

2³ = 8 and 3³ = 27. Since 9 lies between 8 and 27, the tenth digit is 2.

Combine the results:

The cube root of 9261 is 21.

Example 2: Find the Cube Root And Cube of 97336.

Group the digits: 97336=(97) (336))

Use the last digit (6) of the number 336:

From the cube table, the cube root of a number ending in 6 also ends in 6.

The unit digit of the cube root is 6.

Determine the range of cubes for the first group (97):

4³ = 64 and 5³  = 125. Since 97 lies between 64 and 125, the tens digit is 4.

Combine the results:

The cube root of 97336 is 46.

Cube Root And Cube

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