RD Sharma Solutions for Class 8 Chapter 3 - Squares and Square Roots Exercise 3.3

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 obstacles. The RD Sharma Solutions Class 8 are prepared according to the concepts covered in the textbook to understand each and every problems 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 column method, visual method and the diagonal method for squaring a two-digit number.

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1. Find the squares of the following numbers using 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

∴ 25² = 625

Where, it can be expressed as

25² = 25× 25 = 625

(ii) 37

So here, a = 3 and b = 7

∴ 37² = 1369

Where, it can be expressed as

25² = 37× 37 = 1369

(iii) 54

So here, a = 5 and b = 4

∴ 54² = 2916

Where, it can be expressed as

54² = 54 × 54 = 2916

(iv) 71

So here, a = 7 and b = 1

∴ 71² = 4941

Where, it can be expressed as

71² = 71 × 71 = 4941

(v) 96

So here, a = 9 and b = 6

∴ 96² = 9216

Where, it can be expressed as

96² = 96 × 96 = 9216

2. Find the squares of the following numbers using 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 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 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 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 left-most 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 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 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 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 left-most 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 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 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 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 left-most 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 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 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 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 left-most 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 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 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 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 left-most 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 identify (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 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, this exercise 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 in practising and learning each and every concept as it provides solutions to all questions asked in RD Sharma textbook.