# Chapter 7 – Cubes and Cube Roots

The following Topics and Sub-Topics are covered in this chapter and are available on MSVgo:

Introduction

Have you heard the intriguing tale of the two famous mathematicians named S. Ramanujan and Prof. G.H. Hardy? Once Prof. Hardy came to meet him and his taxi number was 1729. Prof. Hardy opined that this is a dull number, while Ramanujan claimed that it is a riveting number as it represents that cube of two numbers in two different ways.

1729 = 1728 + 1 = 123 + 13

1729 = 1000 + 729 = 103 + 93

Since then, this number has been called the Hardy – Ramanujan Number. But have you pondered how Ramanujan knew this? He loved numbers and believed that numbers can be constituted into the form of cube and squares also. This led to the emergence of Cubes and Cube Roots in mathematics.

Cubes and Cube Roots

Now let start with the meaning and introduction to cubes and cube roots. Comprehending the concepts of cubes and cube roots can help you in the long run in solving the various complex mathematical equations and even real-life problems.

#### Cubes

A cube refers to a solid figure consisting of equal sides. In other words, a cube implies a three-dimensional figure that gets curated when six squares of the same length are closed off.

Real-life examples of cubes

Some examples of a cube include square, ice cubes, dice, sugar cubes, tables, and milk crates, etc.

Properties of cubes

The following are the paramount properties of the cube:

1. Cubes of any odd numbers are odd.
2. Cubes of any even number are even.
3. In a cube, any number halt with two will end with eight, and vice versa.
4. The first n numbers of the cube equal to the square of its sum.

#### Cubes Roots

Suppose you have a cardboard box in your hand and have been asked to fill it with the boxes of chocolates.  But how do you find out how many boxes can be put in the cardboard box? Yes, you guessed it right. You will find cube roots of its volume. Any number refers to the factor that gets multiplied thrice to acquire that number, it is called cube roots. The symbol for cube root is  3√.

For Example – 2x2x2 = 8. Cube root of 8 is 2

• Ways to find Cube Roots
1. Prime factorization Method- Follows following steps to find the cube root of any number-
• Step 1- Firstly, try to find the number into prime factors like: 5832=(2×2×2)×(3×3×3)×(3×3×3)
• Step 2- Club the same prime factors into groups.
• Step 3- Hand out one prime number from each group and multiply it. Their product will be cube root.  For example – 5832=3√5832=2×3×3=18
2. Estimation Method- Following steps get followed to find the cube root of any number:
• Step 1- From the given cube number, try to formulate the group of 3 commencing from the rightmost digit of the given number. For instance – 117649– 1st group (649) and the 2nd group (117).
• Step 2- The unit digit of the cube root will get decided by the unit digit of the 1st group. Hence, the cube root unit digit will be 6
• Step 3-  Now discover the cube number between which 2nd group lies and find the smaller number of the digit. Thus, 49 is the cube root of 117649.

#### FAQs

1. What are cubes and cube roots?Cube refers to any three dimensional solid embraced by six square facets with three meetings at each vertex. Cube is considered one of the five platonic solids and is only regular hexahedron. Cube roots of any number allude to the factor that gets multiplied three times to obtain that number.
2. How do you find the cubed root of a cube?Try to identify the integer that multiplied by itself three times and, you will get the cubed root of a cube.
3. What does cube root mean?Cube root is a special number that gets obtained after the multiplication three times of that number to obtain the value. For Example – 4 × 4 × 4 = 64, so the cube root of 64 is 4.
4. What is the cube root for 1000?
The cube root of 1000 = 3√1000
= 10
Explanation-
10×10×10 = 1000
5. Is 75 a perfect cube?
It can be taken a gander at exponentiation that includes the base 75 and the type 3. The term gets articulated as the third intensity of 75 or 75 cubed. 75 is a perfect cube on the grounds that the number is the result of the three equivalent whole numbers 75 or in exponential form. It can further be explained as it tends to get represented as 75 × 75 × 75.
6. What are cube roots used for?
Uses of cube roots are as follows-

•  It gets mostly used to solve any cubic equation.
• When any object has a three-dimensional of a certain volume, cube roots often get employed to find the dimension of the given object.
• In geometry, the cube root is used to identify the length of the side of a cubed.

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Proper cognizance and understanding of the meaning and concepts of cubes and cube roots can help you to solve any problem or equations related to cubes and cube roots. Moreover, this is one of the pivotal chapters in mathematics from the point of view of exams.

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### Middle School Math

• Addition
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