[Maths Class Notes] on Cubes From 1 to 50 Pdf for Exam

The cube of a number (x) is the resultant number when it is multiplied three times. As a result, the cube of the number (x) is [ x^{3}] or x-cubed. Let’s use the number 5 as an example. We already know that 555 equals 125. As a result, 125 is known as the 5th cube.

Cubes are the result of multiplying a number by itself three times.

Let’s take an example: 3 × 3 × 3 = 27, so 27 is a cube.

The whole number is multiplied three times, just as the sides for a cube.

Here are the few starting cube numbers:

1 (=1×1×1)

8 (=2×2×2)

27 (=3×3×3)

64 (=4×4×4)

125 (=5×5×5) … etc.

What is a Perfect Cube?

A perfect cube is a number that may be written as the product of three integers that are all the same or equal. For instance, the number 125 is a perfect cube since 53 = 555 = 125. However, 121 is not a perfect cube because there is no integer that produces 121 when multiplied three times. A perfect cube, in other terms, is a number whose cube root is an integer.

What is a Cubed Number?

Suppose we multiply a whole number (not a fraction) by itself, and then again by itself. The result is found to be a cubed number. For example, 3 x 3 x 3 = 27.

A very easy way to write a cube of 3 is [3^{3}]. It means that three is multiplied by itself three times.

The easiest way to do this calculation is to do the multiplication first (3×3) and then to multiply the answer by the same number; 3 x 3 x 3 = 9 x 3 = 27.

The cube is also known as the number which is multiplied by its square.

[ n^{3}] is equal to [ n times n^{3}], which is equal to n × n × n.

The cube function is the known function x ↦ x3 (which is often denoted as y = [ x^{3}]) that connects a number to its cube value. It is an odd function, as

[ -n^{3} = – n^{3}]

Finding the Cube of a Negative Number

Cube for a negative number will always be shown negative, just as the cube of a positive number will always be a positive outcome.

For Example, -125 = -5 x -5 x- -5 = (25 x -5) = -125.

 

Just like the whole numbers (integers), it is very easy to cube a decimal number. You just have to multiply the given decimal number three times.

Finding the Cube of a Decimal

How to find the cube root of a number?

The prime factorization method can be used to find a number’s cube root. Begin with determining the prime factorization of the given number to determine the cube root. Then, divide the resulting components into groups of three identical elements. Then, to find the result, delete the cube root symbol and multiply the factors. If any factor remains that cannot be divided equally into groups of three, the provided number is not a perfect cube, and the cube root of that integer cannot be found. The resultant number (product) of multiplying a number three times by itself is known as the cube of the original number. It’s called a cube since it’s used to indicate a cube’s volume. To find the cube of a number, multiply it by itself first, then multiply the result by the original number once more.

Cube of a fraction 

The cube of a fraction can be calculated by multiplying it three times, just as the cube of an integer.

Cube of a negative number 

Finding the cube of a negative number follows the same steps as finding the cube of a whole number or fraction. Always remember that a negative number’s cube is always negative, while a positive number’s cube is always positive.

Properties of Cube root

There is one real number y such that [ y^{3} = x] for any real number x. Because the cube function is rising, it does not produce the same output for two different inputs, and it is applicable to all real values. In other words, it’s a one-to-one bijection. After that, we may define a one-to-one inverse function. We can establish a unique cube root of all real numbers for real numbers. The cube root of a negative number is a negative number if this definition is used.

How to find the cube root of a number?

The inverse technique of determining a number’s cube is called cube root. The symbol ∛ is used to represent it.

List of cube roots 

Using the Division Method to Find the Cube Root

Using the division approach to obtain the cube root is akin to using the long division method or the manual square method. From the back to the front, make a set of three-digit numbers. The next step is to determine which number has a cube root that is less than or equal to the supplied integer. Subtract the acquired number from the specified number, then enter the result in the second box. Following this, get the multiplication factor for the next step in the long division procedure, which is done by multiplying the initial number obtained. To find the cube root of a number, repeat the technique described above. When the given integer is not a perfect cube number, this lengthy division method is utilized. It takes a long time to find the cube root of an integer using this method.

Solved Questions 

1. Calculate the Cube 5 and 6.

Solution: To find the cube of 5, we will multiply 5 three times, i.e., 5× 5 ×5 = 125.

To find the cube of 6, we multiply 6 three times, i.e., 6× 6 ×6 = 216. 

2. Find the Cube of (0.06)³.

Solution: 

0.06 can be written as (6/100)³.

Simplifying (6/100)³, we get 

(3/50)³ = (3)³/(50)³

(3 * 3*3)/(50 * 50* 50)

= 27/125000

Hence, the cube of (0.06) is 27/125000.