Cube Root of a Cube Number using Estimation
In this method, we can estimate cube of a perfect cube number using some rule. The steps to estimate cube of perfect cube number are as follows:
Step 1: Take any cube number say 117649 and start making groups of three starting from the rightmost digit of the number.
So 117649 has two groups, and the first group is (649) and the second group is (117).
Step 2: The unit’s digit of the first group (649) will decide the unit digit of the cube root. Since 649 ends with 9, the cube root’s unit digit is 9.
Note: We can use the following table for finding the unit digit of cube root,
Unit digit of Cube Root 1 2 3 4 5 6 7 8 9 Unit digit of its Cube 1 8 7 4 5 6 3 2 9 Step 3: Find the cube of numbers between which the second group lies. The other group is 117.
We know that 403= 64000 i.e., second group for cube of 40 is 64, and 503= 125000 i.e., second group for 50 is 125. As 64 < 117 < 125. Thus, the ten’s digit of the requred number is either 4 or 5 and 50 is the least number with 5 as ten’s digit. Thus, 4 is the ten’s digit of the given number.
So, 49 is cube root of 117649.
Note: For help with second group we can use the following table,
Number 0 10 20 30 40 50 60 70 80 90 Cube 0 1000 8000 27000 64000 125000 216000 343000 512000 729000
Example: Estimate Cube root of number 357911.
Let’s take another cube number, say 175616.
Step 1: Starting from the rightmost digit, group the digits in threes. So, the first group is (616) and the second group is (175).
Step 2: The unit digit of the first group (616) is 6, which corresponds to the unit digit of the cube root 6.
Step 3: Find the cubes of numbers between which the second group lies. We know that 43³ = 79507 and 44³ = 85184. Since 175 is between 79507 and 85184, the tens digit of the required number is 4.
Therefore, cube root of 175616 is 46.
Hardy-Ramanujan Numbers
Numbers like 1729, 4104,13832, etc. are known as Hardy – Ramanujan Numbers because they can be expressed as the sum of two cubes in two different ways. A number n is said to be Hardy-Ramanujan Number if
n = a3 + b3 = c3 + d3
where,
- a, b, c, and d are all distinct positive integers
First four Hardy-Ramanujan numbers are:
- 1729 = 13 + 123 = 93 + 103
- 4104 = 23 + 163 = 93 + 153
- 13832 = 23 + 243 = 183 + 203
- 20683 = 103 + 273 = 193 + 243
If we consider negative integers as well, then 91 becomes the smallest Hardy-Ramanujan Number as it can be expressed as follows:
91 = 63 + (-5)3 = 43 + 33
Note: Number 1729 is also sometimes referred to as the Taxicab Number as it was the number of taxi taken by Dr. Hardy while going to meet Ramanujan at hospital.
Important Maths Related Links:
Cubes and Cube Roots
Cube is a number which we get after multiplying a number 3 times by itself. For example, 125 will be the cube of 5. While, cube root of a number is that number which is multiplied 3 times to get the original number. For example, the cube root of 125 is 5 as if we multiply 5 three times will be 125.
Cube and Cube Roots are fundamental concepts in algebra. The multiplication of a number to itself gives rise to a square and then if we multiply the number by its square the result becomes a cube and the inverse of the cube is the cube root which we will study in this article.
In this article, we will learn about cubes and cube roots and also learn about the methods to find both cubes of a number and cube roots of a number.
Table of Content
- What are Cubes and Cube Roots?
- What is Cube?
- Cubes 1 to 20
- Properties of Cube of Numbers
- Representation of Cube Numbers
- Cube of Negative Numbers
- Cube of Fractions
- Unit Digits in Cube Numbers
- Perfect Cubes
- What is Cube Root?
- Symbol of Cube Root
- Cube Root Formula
- How to Find Cube Root of a Number?
- By Prime Factorization
- Cube Root of a Cube Number using Estimation
- Hardy-Ramanujan Numbers
- Cube and Cube Roots Class 8 Worksheet
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