Syllabus of Class 8 Mathematics
1.Rational Numbers2.Linear Equations in one variable3.Understanding Quadrilaterals4.Practical Geometry5.Data Handling6.Squares Square Roots7.Cube and Cube Roots8.Comparing Quantities9.Algebraic Expressions and Identities10.Visualising Solid Shapes11.Mensuration12.Exponents and Powers13.Direct and Inverse Properties14.Factorisation15.Introduction to Graph16.Playing with Numbers
Revision Notes on Cubes and Cube Roots
Hardy-Ramanujan Number
The number which can be expressed as the sum of two cubes in different ways is said to be a Hardy – Ramanujan number.
1729 = 1728 + 1 = 123 + 13
1729 = 1000 + 729 = 103 + 93
As 1729 is the smallest such type of number so it is called the smallest Hardy-Ramanujan number. There is infinite such type of numbers. Like- 4104 (2, 16; 9, 15), 13832 (18, 20; 2, 24), etc.
Cubes
Cube is a 3-dimensional figure with all equal sides. If one cube has all the equal sides of 1 cm then how many such cubes are needed to make a new cube of side 2 cm?
8 such cubes are needed, and what if we need to make a cube of side 3 cm with the cubes of side 1 cm? The numbers 1, 8, 27 ...etc can be shown below in the cube.
These are known as perfect cubes or cube numbers. This shows that we got the cube numbers by multiplying the number three times by itself.
Cubes of Some Natural Numbers
Number Cubes Numbers Cubes 1 13 = 1 11 113 = 1331 2 23 = 8 12 123 = 1728 3 33 = 27 13 133 = 2197 4 43 = 64 14 143 = 2744 5 53 = 125 15 153 = 3375 6 63 = 216 16 163 = 4096 7 73 = 343 17 173 = 4913 8 83 = 512 18 183 = 5832 9 93 = 729 19 193 = 6859 10 103 = 1000 20 203 = 8000
| Number | Cubes | Numbers | Cubes |
| 1 | 13 = 1 | 11 | 113 = 1331 |
| 2 | 23 = 8 | 12 | 123 = 1728 |
| 3 | 33 = 27 | 13 | 133 = 2197 |
| 4 | 43 = 64 | 14 | 143 = 2744 |
| 5 | 53 = 125 | 15 | 153 = 3375 |
| 6 | 63 = 216 | 16 | 163 = 4096 |
| 7 | 73 = 343 | 17 | 173 = 4913 |
| 8 | 83 = 512 | 18 | 183 = 5832 |
| 9 | 93 = 729 | 19 | 193 = 6859 |
| 10 | 103 = 1000 | 20 | 203 = 8000 |
Revision Notes on Cubes and Cube Roots
Hardy-Ramanujan Number
The number which can be expressed as the sum of two cubes in different ways is said to be a Hardy – Ramanujan number.
1729 = 1728 + 1 = 123 + 13
1729 = 1000 + 729 = 103 + 93
As 1729 is the smallest such type of number so it is called the smallest Hardy-Ramanujan number. There is infinite such type of numbers. Like- 4104 (2, 16; 9, 15), 13832 (18, 20; 2, 24), etc.
The number which can be expressed as the sum of two cubes in different ways is said to be a Hardy – Ramanujan number.
1729 = 1728 + 1 = 123 + 13
1729 = 1000 + 729 = 103 + 93
As 1729 is the smallest such type of number so it is called the smallest Hardy-Ramanujan number. There is infinite such type of numbers. Like- 4104 (2, 16; 9, 15), 13832 (18, 20; 2, 24), etc.
Cubes
Cube is a 3-dimensional figure with all equal sides. If one cube has all the equal sides of 1 cm then how many such cubes are needed to make a new cube of side 2 cm?
8 such cubes are needed, and what if we need to make a cube of side 3 cm with the cubes of side 1 cm? The numbers 1, 8, 27 ...etc can be shown below in the cube.
These are known as perfect cubes or cube numbers. This shows that we got the cube numbers by multiplying the number three times by itself.
Cube is a 3-dimensional figure with all equal sides. If one cube has all the equal sides of 1 cm then how many such cubes are needed to make a new cube of side 2 cm?
8 such cubes are needed, and what if we need to make a cube of side 3 cm with the cubes of side 1 cm? The numbers 1, 8, 27 ...etc can be shown below in the cube.
These are known as perfect cubes or cube numbers. This shows that we got the cube numbers by multiplying the number three times by itself.
Cubes of Some Natural Numbers
Number Cubes Numbers Cubes 1 13 = 1 11 113 = 1331 2 23 = 8 12 123 = 1728 3 33 = 27 13 133 = 2197 4 43 = 64 14 143 = 2744 5 53 = 125 15 153 = 3375 6 63 = 216 16 163 = 4096 7 73 = 343 17 173 = 4913 8 83 = 512 18 183 = 5832 9 93 = 729 19 193 = 6859 10 103 = 1000 20 203 = 8000
This table shows that
There are only 10 perfect cubes between 1-1000.
The cube of an even number is also even.
The cube of an odd number is also an odd number.
| Number | Cubes | Numbers | Cubes |
| 1 | 13 = 1 | 11 | 113 = 1331 |
| 2 | 23 = 8 | 12 | 123 = 1728 |
| 3 | 33 = 27 | 13 | 133 = 2197 |
| 4 | 43 = 64 | 14 | 143 = 2744 |
| 5 | 53 = 125 | 15 | 153 = 3375 |
| 6 | 63 = 216 | 16 | 163 = 4096 |
| 7 | 73 = 343 | 17 | 173 = 4913 |
| 8 | 83 = 512 | 18 | 183 = 5832 |
| 9 | 93 = 729 | 19 | 193 = 6859 |
| 10 | 103 = 1000 | 20 | 203 = 8000 |
This table shows that
There are only 10 perfect cubes between 1-1000.
The cube of an even number is also even.
The cube of an odd number is also an odd number.
One’s digit of the Cubes
One’s digit of the Cubes of a number having a particular number at the end will always remain same. Let’s see in the following table:
Unit’s digit of number Last digit of its cube number Example 1 1 113 = 1331, 213 = 9261, etc. 2 8 23 = 8, 123 = 1728, 323 = 32768, etc. 3 7 133 = 2197, 533 = 148877, etc. 4 4 243 = 13824, 743 = 405224, etc. 5 5 153 = 3375, 253 = 15625, etc. 6 6 63 = 216, 263 = 17576,etc. 7 3 173 = 4913, 373 = 50653,etc. 8 2 83 = 512, 183 = 5832, etc. 9 9 193 = 6859, 393 = 59319, etc. 10 20 103 = 1000, 203 = 8000, etc.
One’s digit of the Cubes of a number having a particular number at the end will always remain same. Let’s see in the following table:
| Unit’s digit of number | Last digit of its cube number | Example |
| 1 | 1 | 113 = 1331, 213 = 9261, etc. |
| 2 | 8 | 23 = 8, 123 = 1728, 323 = 32768, etc. |
| 3 | 7 | 133 = 2197, 533 = 148877, etc. |
| 4 | 4 | 243 = 13824, 743 = 405224, etc. |
| 5 | 5 | 153 = 3375, 253 = 15625, etc. |
| 6 | 6 | 63 = 216, 263 = 17576,etc. |
| 7 | 3 | 173 = 4913, 373 = 50653,etc. |
| 8 | 2 | 83 = 512, 183 = 5832, etc. |
| 9 | 9 | 193 = 6859, 393 = 59319, etc. |
| 10 | 20 | 103 = 1000, 203 = 8000, etc. |
Some Interesting Patterns
1. Adding Consecutive Odd Numbers
This shows that if we add the consecutive odd numbers then we get the cube of the next number.
2. Cubes and their Prime Factors
Prime factorization of a number is done by finding the prime factors of the number and then pairing it in the group of three. If all the prime factors are in the pair of three then the number is a perfect cube.
Example
Calculate the cube root of 13824 by using prime factorization method.
Solution
First of all write the prime factors of the given number then pair them in the group of three.
Since all the factors are in the pair of three the number 13824 is a perfect cube.
1. Adding Consecutive Odd Numbers
This shows that if we add the consecutive odd numbers then we get the cube of the next number.
2. Cubes and their Prime Factors
Prime factorization of a number is done by finding the prime factors of the number and then pairing it in the group of three. If all the prime factors are in the pair of three then the number is a perfect cube.
Example
Calculate the cube root of 13824 by using prime factorization method.
Solution
First of all write the prime factors of the given number then pair them in the group of three.
Since all the factors are in the pair of three the number 13824 is a perfect cube.
Smallest Multiple that is a Perfect Cube
As we have seen that the group of three prime factors makes a number perfect cube, so to make a number perfect cube we need to multiply it with the smallest multiple of that number.
Example
Check whether 1188 is a perfect cube or not. If not then which smallest natural number should be multiplied to 1188 to make it a perfect cube?
Solution
1188 = 2 × 2 × 3 × 3 × 3 × 11
This shows that the prime numbers 2 and 11 are not in the groups of three. So, 1188 is not a perfect cube
To make it a perfect cube we need to multiply it with 2 × 11 × 11 = 242, so, it will make the pair of 2, 3 and 11.
Hence the smallest natural number by which 1188 should be multiplied to make it a perfect cube is 242.
And the resulting perfect cube is 1188 × 242 = 287496 ( = 663).
As we have seen that the group of three prime factors makes a number perfect cube, so to make a number perfect cube we need to multiply it with the smallest multiple of that number.
Example
Check whether 1188 is a perfect cube or not. If not then which smallest natural number should be multiplied to 1188 to make it a perfect cube?
Solution
1188 = 2 × 2 × 3 × 3 × 3 × 11
This shows that the prime numbers 2 and 11 are not in the groups of three. So, 1188 is not a perfect cube
To make it a perfect cube we need to multiply it with 2 × 11 × 11 = 242, so, it will make the pair of 2, 3 and 11.
Hence the smallest natural number by which 1188 should be multiplied to make it a perfect cube is 242.
And the resulting perfect cube is 1188 × 242 = 287496 ( = 663).
Cube Roots
Finding cube root is the inverse operation of finding the cube.
If 33 =27 then cube root of 27 is 3.
We write it as ∛27 = 3
Symbol of the Cube Root
Some of the cube roots are:
Statement Inference Statement Inference 13 = 1 ∛1 = 1 63 = 216 ∛216 = ∛63 = 6 23 = 8 ∛8 = ∛23 = 2 73 = 343 ∛343 = ∛73 = 7 33 = 27 ∛27 = ∛33 = 3 83 = 512 ∛512 = ∛83 = 8 43 = 64 ∛64 = ∛43 = 4 93 = 729 ∛729 = ∛93 = 9 53 = 125 ∛125 = ∛53 = 5 103 = 1000 ∛1000 = ∛103 = 10
Finding cube root is the inverse operation of finding the cube.
If 33 =27 then cube root of 27 is 3.
We write it as ∛27 = 3
Symbol of the Cube Root
Some of the cube roots are:
| Statement | Inference | Statement | Inference |
| 13 = 1 | ∛1 = 1 | 63 = 216 | ∛216 = ∛63 = 6 |
| 23 = 8 | ∛8 = ∛23 = 2 | 73 = 343 | ∛343 = ∛73 = 7 |
| 33 = 27 | ∛27 = ∛33 = 3 | 83 = 512 | ∛512 = ∛83 = 8 |
| 43 = 64 | ∛64 = ∛43 = 4 | 93 = 729 | ∛729 = ∛93 = 9 |
| 53 = 125 | ∛125 = ∛53 = 5 | 103 = 1000 | ∛1000 = ∛103 = 10 |
Method of finding a Cube Root
There are two methods of finding a cube root
1. Prime Factorization Method
Step 1: Write the prime factors of the given number.
Step 2: Make the pair of three if possible.
Step 3: Then replace them with a single digit.
Step 4: Multiply these single digits to find the cube root.
Example
Find the cube root of 15625 by the prime factorization method.
2. Estimation Method
This method is based on the estimation. Let's take the above example.
Step 1: If 15625 is the number then make the group of three digits starting from the right.
15 625
Step2: Here 625 is the first group which tells us the unit’s digit of the cube root. As the number is ending with 5 and we know that 5 comes at the unit’s place of a number only when its cube root ends in 5.
So the unit place is 5.
Step 3: Now take the other group, i.e., 15. Cube of 2 is 8 and a cube of 3 is 27. 15 lie between 8 and 27. The number which is smaller among 2 and 3 is 2. The one’s place of 2 is 2 itself. Take 2 as ten’s place of the cube root of 15625. Thus,
There are two methods of finding a cube root
1. Prime Factorization Method
Step 1: Write the prime factors of the given number.
Step 2: Make the pair of three if possible.
Step 3: Then replace them with a single digit.
Step 4: Multiply these single digits to find the cube root.
Example
Find the cube root of 15625 by the prime factorization method.
2. Estimation Method
This method is based on the estimation. Let's take the above example.
Step 1: If 15625 is the number then make the group of three digits starting from the right.
15 625
Step2: Here 625 is the first group which tells us the unit’s digit of the cube root. As the number is ending with 5 and we know that 5 comes at the unit’s place of a number only when its cube root ends in 5.
So the unit place is 5.
Step 3: Now take the other group, i.e., 15. Cube of 2 is 8 and a cube of 3 is 27. 15 lie between 8 and 27. The number which is smaller among 2 and 3 is 2. The one’s place of 2 is 2 itself. Take 2 as ten’s place of the cube root of 15625. Thus,
Contact Us
Welcome to www.fuzymail.co.in
Please email us if you have any queries about the site, advertising, or anything else.
We will revert you as soon as possible...
Thank you for contacting us!
Have a great day
.png)
No comments:
Post a Comment
thanks for your feedback