RD Sharma Solutions for Class 8 Maths Exercise 3.3 Chapter 3, Squares and Square Roots, are provided here. The questions present in this exercise have been solved by BYJU’S subject experts, and this will help students in solving the questions without any difficulty. The RD Sharma Solutions Class 8 are prepared according to the concepts covered in the textbook to understand each and every problem thoroughly. Exercise 3.3 of RD Sharma Solutions for Class 8 helps students to understand the concepts of some shortcuts to find squares which include the column method, the visual method and the diagonal method for squaring a two-digit number.
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EXERCISE 3.3 PAGE NO: 3.32
1. Find the squares of the following numbers using the column method. Verify the result by finding the square using the usual multiplication:
(i) 25
(ii) 37
(iii) 54
(iv) 71
(v) 96
Solution:
(i) 25
So here, a = 2 and b = 5
| Column I | Column II | Column III |
| a2
4 +2 6 |
2ab
20 +2 22 |
b2
25 |
| 6 | 2 | 5 |
∴ 252 = 625
Where, it can be expressed as
252 = 25× 25 = 625
(ii) 37
So here, a = 3 and b = 7
| Column I | Column II | Column III |
| a2
9 +4 13 |
2ab
42 +4 46 |
b2
49 |
| 13 | 6 | 9 |
∴ 372 = 1369
Where, it can be expressed as
252 = 37× 37 = 1369
(iii) 54
So here, a = 5 and b = 4
| Column I | Column II | Column III |
| a2
25 +4 29 |
2ab
40 +1 41 |
b2
16 |
| 29 | 1 | 6 |
∴ 542 = 2916
Where, it can be expressed as
542 = 54 × 54 = 2916
(iv) 71
So here, a = 7 and b = 1
| Column I | Column II | Column III |
| a2
49 +1 50 |
2ab
14 +0 14 |
b2
01 |
| 50 | 4 | 1 |
∴ 712 = 5041
Where, it can be expressed as
712 = 71 × 71 = 5041
(v) 96
So here, a = 9 and b = 6
| Column I | Column II | Column III |
| a2
81 +11 92 |
2ab
108 +3 111 |
b2
36 |
| 92 | 1 | 6 |
∴ 962 = 9216
Where, it can be expressed as
962 = 96 × 96 = 9216
2. Find the squares of the following numbers using the diagonal method:
(i) 98
(ii) 273
(iii) 348
(iv) 295
(v) 171
Solution:
(i) 98
Step 1: Obtain the number and count the number of digits in it. Let there be n digits in the number to be squared.
Step 2: Draw a square and divide it into n2 sub-squares of the same size by drawing (n – 1) horizontal and (n – 1) vertical lines.
Step 3: Draw the diagonals of each sub-square.
Step 4: Write the digits of the number to be squared along the left vertical side sand top horizontal side of the squares.
Step 5: Multiply each digit on the left of the square with each digit on top of the column one by one. Write the units digit of the product below the diagonal and the tens digit above the diagonal of the corresponding sub-square.
Step 6: Starting below the lowest diagonal, sum the digits along the diagonals so obtained. Write the units digit of the sum and take carry, the tens digit (if any) to the diagonal above.
Step 7: Obtain the required square by writing the digits from the leftmost side.
∴ 982 = 9604
(ii) 273
Step 1: Obtain the number and count the number of digits in it. Let there be n digits in the number to be squared.
Step 2: Draw a square and divide it into n2 sub-squares of the same size by drawing (n – 1) horizontal and (n – 1) vertical lines.
Step 3: Draw the diagonals of each sub-square.
Step 4: Write the digits of the number to be squared along the left vertical side and the top horizontal side of the squares.
Step 5: Multiply each digit on the left of the square with each digit on top of the column one by one. Write the units digit of the product below the diagonal and the tens digit above the diagonal of the corresponding sub-square.
Step 6: Starting below the lowest diagonal, sum the digits along the diagonals so obtained. Write the units digit of the sum and take carry, the tens digit (if any) to the diagonal above.
Step 7: Obtain the required square by writing the digits from the leftmost side.
∴ 2732 = 74529
(iii) 348
Step 1: Obtain the number and count the number of digits in it. Let there be n digits in the number to be squared.
Step 2: Draw a square and divide it into n2 sub-squares of the same size by drawing (n – 1) horizontal and (n – 1) vertical lines.
Step 3: Draw the diagonals of each sub-square.
Step 4: Write the digits of the number to be squared along the left vertical side sand top horizontal side of the squares.
Step 5: Multiply each digit on the left of the square with each digit on top of the column one by one. Write the units digit of the product below the diagonal and the tens digit above the diagonal of the corresponding sub-square.
Step 6: Starting below the lowest diagonal, sum the digits along the diagonals so obtained. Write the units digit of the sum and take carry, the tens digit (if any) to the diagonal above.
Step 7: Obtain the required square by writing the digits from the leftmost side.
∴ 3482 = 121104
(iv) 295
Step 1: Obtain the number and count the number of digits in it. Let there be n digits in the number to be squared.
Step 2: Draw a square and divide it into n2 sub-squares of the same size by drawing (n – 1) horizontal and (n – 1) vertical lines.
Step 3: Draw the diagonals of each sub-square.
Step 4: Write the digits of the number to be squared along the left vertical side and the top horizontal side of the squares.
Step 5: Multiply each digit on the left of the square with each digit on top of the column one by one. Write the units digit of the product below the diagonal and the tens digit above the diagonal of the corresponding sub-square.
Step 6: Starting below the lowest diagonal, sum the digits along the diagonals so obtained. Write the units digit of the sum and take carry, the tens digit (if any) to the diagonal above.
Step 7: Obtain the required square by writing the digits from the leftmost side.
∴ 2952 = 87025
(v) 171
Step 1: Obtain the number and count the number of digits in it. Let there be n digits in the number to be squared.
Step 2: Draw a square and divide it into n2 sub-squares of the same size by drawing (n – 1) horizontal and (n – 1) vertical lines.
Step 3: Draw the diagonals of each sub-square.
Step 4: Write the digits of the number to be squared along the left vertical side sand top horizontal side of the squares.
Step 5: Multiply each digit on the left of the square with each digit on top of the column one by one. Write the units digit of the product below the diagonal and the tens digit above the diagonal of the corresponding sub-square.
Step 6: Starting below the lowest diagonal, sum the digits along the diagonals so obtained. Write the units digit of the sum and take carry, the tens digit (if any) to the diagonal above.
Step 7: Obtain the required square by writing the digits from the leftmost side.
∴ 1712 = 29241
3. Find the squares of the following numbers:
(i) 127
(ii) 503
(iii) 450
(iv) 862
(v) 265
Solution:
(i) 127
1272 = 127 × 127 = 16129
(ii) 503
5032 = 503 × 503 = 253009
(iii) 450
4502 = 450 × 450 = 203401
(iv) 862
8622 = 862 × 862 = 743044
(v) 265
2652 = 265 × 265 = 70225
4. Find the squares of the following numbers:
(i) 425
(ii) 575
(iii)405
(iv) 205
(v) 95
(vi) 745
(vii) 512
(viii) 995
Solution:
(i)425
4252 = 425 × 425 = 180625
(ii) 575
5752 = 575 × 575 = 330625
(iii)405
4052 = 405 × 405 = 164025
(iv) 205
2052 = 205 × 205 = 42025
(v) 95
952 = 95 × 95 = 9025
(vi) 745
7452 = 745 × 745 = 555025
(vii) 512
5122 = 512 × 512 = 262144
(viii) 995
9952 = 995 × 995 = 990025
5. Find the squares of the following numbers using the identity (a+b) 2= a2+2ab+b2:
(i) 405
(ii) 510
(iii) 1001
(iv) 209
(v) 605
Solution:
(i) 405
We know, (a+b) 2= a2+2ab+b2
405 = (400+5)2
= (400)2 + 52 + 2 (400) (5)
= 160000 + 25 + 4000
= 164025
(ii) 510
We know, (a+b) 2= a2+2ab+b2
510 = (500+10)2
= (500)2 + 102 + 2 (500) (10)
= 250000 + 100 + 10000
= 260100
(iii) 1001
We know, (a+b) 2= a2+2ab+b2
1001 = (1000+1)2
= (1000)2 + 12 + 2 (1000) (1)
= 1000000 + 1 + 2000
= 1002001
(iv) 209
We know, (a+b) 2= a2+2ab+b2
209 = (200+9)2
= (200)2 + 92 + 2 (200) (9)
= 40000 + 81 + 3600
= 43681
(v) 605
We know, (a+b) 2= a2+2ab+b2
605 = (600+5)2
= (600)2 + 52 + 2 (600) (5)
= 360000 + 25 + 6000
= 366025
6. Find the squares of the following numbers using the identity (a-b) 2= a2-2ab+b2
(i) 395
(ii) 995
(iii)495
(iv) 498
(v) 99
(vi) 999
(vii)599
Solution:
(i) 395
We know, (a-b) 2= a2-2ab+b2
395 = (400-5)2
= (400)2 + 52 – 2 (400) (5)
= 160000 + 25 – 4000
= 156025
(ii) 995
We know, (a-b) 2= a2-2ab+b2
995 = (1000-5)2
= (1000)2 + 52 – 2 (1000) (5)
= 1000000 + 25 – 10000
= 990025
(iii) 495
We know, (a-b) 2= a2-2ab+b2
495 = (500-5)2
= (500)2 + 52 – 2 (500) (5)
= 250000 + 25 – 5000
= 245025
(iv) 498
We know, (a-b) 2= a2-2ab+b2
498 = (500-2)2
= (500)2 + 22 – 2 (500) (2)
= 250000 + 4 – 2000
= 248004
(v) 99
We know, (a-b) 2= a2-2ab+b2
99 = (100-1)2
= (100)2 + 12 – 2 (100) (1)
= 10000 + 1 – 200
= 9801
(vi) 999
We know, (a-b) 2= a2-2ab+b2
999 = (1000-1)2
= (1000)2 + 12 – 2 (1000) (1)
= 1000000 + 1 – 2000
= 998001
(vii) 599
We know, (a-b) 2= a2-2ab+b2
599 = (600-1)2
= (600)2 + 12 – 2 (600) (1)
= 360000 + 1 – 1200
= 358801
7. Find the squares of the following numbers by the visual method:
(i) 52
(ii) 95
(iii) 505
(iv) 702
(v) 99
Solution:
(i) 52
We know, (a+b) 2= a2+2ab+b2
52 = (50+2)2
= (50)2 + 22 + 2 (50) (2)
= 2500 + 4 + 200
= 2704
(ii) 95
We know, (a-b) 2= a2-2ab+b2
95 = (100-5)2
= (100)2 + 52 – 2 (100) (5)
= 10000 + 25 – 1000
= 9025
(iii) 505
We know, (a+b) 2= a2+2ab+b2
505 = (500+5)2
= (500)2 + 52 + 2 (500) (5)
= 250000 + 25 + 5000
= 255025
(iv) 702
We know, (a+b) 2= a2+2ab+b2
702 = (700+2)2
= (700)2 + 22 + 2 (700) (2)
= 490000 + 4 + 2800
= 492804
(v) 99
We know, (a-b) 2= a2-2ab+b2
99 = (100-1)2
= (100)2 + 12 – 2 (100) (1)
= 10000 + 1 – 200
= 9801
RD Sharma Solutions for Class 8 Maths Exercise 3.3 Chapter 3 – Squares and Square Roots
Exercise 3.3 of RD Sharma Solutions for Chapter 3 Squares and Square Roots, mainly deals with the basic concepts related to methods for squaring a two-digit number. Here are a few methods:
Column method for squaring a two-digit number
- Visual method for squaring a two-digit number
- Diagonal method for squaring a two-digit number
The RD Sharma Solutions can help students practise and learn each and every concept as it provides solutions to all questions asked in the RD Sharma textbook.
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