Cube Root from 1 to 30

Not all numbers between 1 and 30 are perfect cubes, so their cube roots are irrational and are generally expressed as decimals. There are three perfect cubes between 1 to 30. These numbers are 1, 8, and 27, and their cube roots are whole numbers 1, 2, and 3, respectively. 

The cube root chart works as an educational aid for students. It consists of all the cube roots of numbers between 1 to 30; it helps solve problems more quickly.

Cube roots from 1 to 30 are useful in problem-solving for number operations. Memorizing these roots helps students gain efficiency in solving questions. Below is a list of all cube roots between 1 and 30.

Cube roots from 1 to 10

The cube roots between 1 and 10, including whole numbers as well as decimals, are given below:
 

Cube roots from 11 to 20

As the natural numbers progress, their cube roots start to complicate. However, the process of calculating cube roots remains the same. These irrational cube roots can be rounded off to a few decimal places.
 

Cube roots from 21 to 30
 

3 is the only whole cube root within this range. All other cube roots from 21 to 30 are irrational with higher decimal values. Do not worry about the large number of decimal values; you can round them off to a few places after the decimal point.

Perfect cubes are numbers formed by multiplying an integer by itself 3 times. Their cube roots are also whole numbers. Between 1 and 30, the only perfect cubes that exist are 1, 8, and 27, with 1, 2, and 3 being their respective cube roots.

A non-perfect cube has a cube root that is not a whole number. Its cube root is always an irrational number. This applies to all integers between 1 and 30 except for 1, 8, and 27.

The cube roots from 1 to 30 are calculated:

• By Prime Factorization Method
   
• By Estimation Method

The prime factorization method helps us find perfect cube roots. Follow these steps for calculating the cube roots:

Step 1: Compute the prime factors of the number.

Step 2: Group the prime factors in triples.

Step 3: After forming triplets, take only one value from each group to find the cube root.

Let's solve an example using these steps:

Question: Find the cube root of 8

Solution: Prime factorization of 8, 8 = 2 × 2 × 2 

We have one group of triples 2³

So, 2 is the cube root of 8.

The estimation method is used for non-perfect cubes so that we can find the approximate root between two nearly perfect cubes. To do so,

Step 1: Find two perfect cubes nearest to the number. The number should lie between them.

Step 2: Make estimations based on how close the numbers are.

Step 3: Refine the estimation by trying different values through multiplication.

For example, to estimate 320

First, check for the nearest perfect cubes

23 = 3 and 33 = 27 

 320 lies between 2 and 3
 
Since 20 is closer to 27, the estimate will be between 2.7 and 2.8

Now, try 2.73 = 19.683

2.723 = 20.138

2.713= 19.909

Upon refining these estimates further, we get

∛20 ≈ 2.72 as the answer.

Now that we know the two methods of finding cube roots, let's focus on the rules that must be followed while finding cube roots:

Rule 1: Identifying a Perfect Cube

A perfect cube always has a whole number as its cube root. These are also called exact cubes. Between 1 - 30, the perfect cubes are 1, 8, and 27.

Rule 2: Estimating Cube Roots of Non-Perfect Cubes

For numbers that aren't perfect cubes, their cube roots can be estimated by locating the two nearest perfect cubes.

Rule 3: Cube Root Formula

The cube root of a number y is a number x such that: x = y^1/3
Let's say we want to find the cube root of 27, then
x = 27^1/3 = 3

Rule 4: Important Properties of Cube Roots

• A negative number will have a negative cube. 
  For example, ∛(−8) = −2 because (−2)³ = −8

• When multiplying or dividing cube roots, you can separate the numbers under the root.
  For example,
  ∛(8 × 27) = ∛8 × ∛27 = 2 × 3 = 6 
  Similarly, for division, ∛(27 ÷ 3) = ∛27 ÷ ∛3 = 3 ÷ 1.44 ≈ 2.08

Picking up a few tips and tricks can help students understand the topic more efficiently. Some helpful tips are:

• Memorize the perfect cubes between 1-30There are only 3 perfect cubes between 1-30, and memorizing them makes calculations like estimation of cube roots easier. So remember that 1, 8, and 27 are perfect cubes.

• Use trial cubing for closer estimates.

• Negative numbers can also be cube roots. All negative cubes have negative cube roots and vice versa.

• Irrational number: Numbers that cannot be written as simple fractions or whole numbers are irrational numbers.

• Trial cubing: The method of trying different numbers by cubing them to find a cube root is known as trial cubing.

• Exact cube: Exact cube is just another synonym for a perfect cube.