### Question 1. Find the square of the following numbers using column method. Verify the squares using the usual multiplication.

### (i) 25

**Solution:**

Here, we will break 25 in as between one’s and ten’s position as a = 2 and b = 5. Now,

Step 1: Make 3 columns and write the values of a², 2ab, b² in these columns.

Column 1Column 2Column 3a ^{2}2ab

b ^{2}4 20 25 Step 2: Underline the unit digit of b² and add its ten’s digit(if any) to 2ab.

Column 1Column 2Column 3a ^{2}2ab

b ^{2}4 20 + 2 2 522 Step 3: Now Underline the unit digit of column 2 and add its ten’s digit(if any) to a²

Column 1Column 2Column 3a ^{2}2ab

b ^{2}4+2 20+2 2 56 2 2Step 4: Now underline the number in column 1.

Column 1Column 2Column 3a ^{2}2ab

b ^{2}4+2 20+2 2 562 2Step 5: Now write the underlined digits respectively as the square.

25

^{2}= 625Now using multiplication,

25

^{2}= 25 x 25 = 625Because we have the same result in both the methods hence our result is verified.

### (ii) 37

**Solution:**

Here, a = 3, b = 7

Column 1Column 2Column 3a ^{2}2ab

b ^{2}9 + 4 42 + 4 4 9134 6Now write the underlined digits respectively as the square.

37

^{2}= 1369Now using multiplication,

37

^{2}= 37 x 37 = 1369Because we have the same result in both the methods hence our result is verified.

### (iii) 54

**Solution:**

Here, a = 5, b = 4.

Column 1Column 2Column 3a ^{2}2ab

b ^{2}25+4 40+1 1 6294 1Now write the underlined digits respectively as the square.

54

^{2}= 2916Now using multiplication,

54

^{2 }= 54 x 54 = 2916Because we have the same result in both the methods hence our result is verified.

### (iv) 71

**Solution:**

Here, a = 7, b = 1.

Column 1Column 2Column 3a ^{2}2ab

b ^{2}49 + 1 14 + 0 1501 4Now write the underlined digits respectively as the square.

71

^{2}= 5041Now using multiplication,

71

^{2}= 71 x 71 = 5041Because we have the same result in both the methods hence our result is verified.

### (v) 96

**Solution:**

Here, a = 9, b = 6.

Column 1Column 2Column 3a ^{2}2ab

b ^{2}81 + 11 108 + 3 3 69211 3Now write the underlined digits respectively as the square.

96

^{2}= 9216Now using multiplication,

96

^{2}= 96 x 96 = 9216.Because we have the same result in both the methods hence our result is verified.

### Question 2. Find the squares of the following numbers using the diagonal method:

### (i) 98

**Solution:**

Because, 98² = 9604

Draw a square table with equal no. of rows and columns as the no. of digits are.

Now divide each block in two parts.

Now, write the digits as depicted, and we have to store the values as provided in each block

Now Store the values as shown and add them as per the subdivision we have made, i.e., 4, 2 + 6 + 2, 7 + 1 + 7, 8 and the previous carry(i.e., take only one unit’s digit and transfer the other as carry).

Now, write the underlined(unit’s digit), as the square of the number

98

^{2}= 9604

### (ii) 273

**Solution:**

Because, 273

^{2}= 74529Now, write the underlined(unit’s digit), as the square of the number

273

^{2}= 74529

### (iii) 348

**Solution:**

Because, 348

^{2}= 121104Now, write the unit’s digit as the square of 348,

348

^{2 }= 121104

### (iv) 295

**Solution:**

Because, 295

^{2}= 87025Now, write each one’s digit as the square of 295,

295

^{2}= 87025

### (v) 171

**Solution:**

Because, 171

^{2}= 29241Now, write each one’s digit as the square of 171,

171

^{2}= 29241

### Question 3. Find the square of the following numbers:

### (i) 127

**Solution:**

Here let’s take a = 120, b = 7

127

^{2}= (120 + 7)^{2 }= 120^{2}+ (2 x 120 x 7) + 7^{2}= (120 x 120) + 1680 + (7 x 7)

= 14400 + 1680 + 49

= 16129

Alternatively:We could also take a = 100, b = 27,

127

^{2}= 10000 + 5400 + 729 = 16129Thus, 127

^{2}= 16129

### (ii) 503

**Solution:**

Here, let’s take a = 500, b = 3.

503

^{2}= (500 + 3)^{2}= 500^{2}+ (2 x 500 x 3) + 3^{2}= 250000 + 3000 + 9

= 253009

Alternatively:503

^{2}= 503 x 503 = 253009Thus, 503

^{2}= 253009

### (iii) 451

**Solution:**

Here, let’s take a = 400, b = 51.

451

^{2}= (400 + 51)^{2}= 400^{2}+ (2 x 400 x 51) + 51^{2}= 160000 + 40800 + 2601

= 203401

Alternatively:451² = 451 x 451 = 203401

451² = 203401

### (iv) 862

**Solution:**

Here, let’s take a = 800, b = 62.

862

^{2}= (800 + 62)^{2}= 800^{2}+ (2 x 800 x 62) + 62^{2}= 640000 + 99200 + 3844

= 743044

Alternatively:862² = 862 x 862 = 743044

862² = 743044

### (v) 265

**Solution:**

Here, let’s take a = 200, b = 65.

265

^{2}= (200 + 65)^{2}= 200^{2}+ (2 x 200 x 65) + 65^{2}= 40000 + 26000 + 4225

= 70225

Alternatively:265² = 265 x 265 = 70225

265² = 70225