Exercise 5.4

Find the square root of each of the following numbers by Division method.

(i) 2304

Instructions

Find the square root using long division method

Starting from right place a bar on top of every pair of digits. If a single digit remains on the left, add a bar to that as well. For eg: 23 04
From the left, find the square which is less than or equal to the digit. Here we have, (42 < 23 < 52). Here, 4 becomes the first digit of the quotient and we get a remainder of when subtracting 42 from 23.
Bring down the number under the next bar (i.e. 04) to the right of the remainder. We now get the new dividend i.e.
In the divisor put the first digit as double the quotient and place a blank its right. i.e. _
Put the largest possible digit to fill the blank which will also become the new digit in the quotient, such that when the new divisor is multiplied to the new quotient the product is less than or equal to the dividend.
As × = 704, we can choose the digit to be filled in the blank as .
Since, the remainder is and there are no more digits left in the given number, we get: 2304 =
We have found the square root of 2304.

(ii) 4489

Instructions

Finding the square root using long division method

Starting from right place a bar on top of every pair of digits. If a single digit remains on the left, add a bar to that as well. For eg: 44 89
From the left, find the square which is less than or equal to the digit. Here we have, (62 < 44 < 72). Here, becomes the first digit of the quotient and we get a remainder of when subtracting 62 from 44.
Bring down the number under the next bar (i.e. 89) to the right of the remainder. We now get the new dividend i.e.
In the divisor put the first digit as double the quotient and place a blank its right. i.e. _
Put the largest possible digit to fill the blank which will also become the new digit in the quotient, such that when the new divisor is multiplied to the new quotient the product is less than or equal to the dividend.
As × = 889, we can choose the digit to be filled in the blank as .
Since, the remainder is and there are no more digits left in the given number, we get: 4489 =
We have found the square root of 4489.

(iii) 3481

Instructions

Finding square root using long division method

Starting from right place a bar on top of every pair of digits. If a single digit remains on the left, add a bar to that as well. For eg: 34 81
From the left, find the square which is less than or equal to the digit. Here we have, (52 < 34 < 62). Here, 5 becomes the first digit of the quotient and we get a remainder of when subtracting 52 from 34.
Bring down the number under the next bar (i.e. 81) to the right of the remainder. We now get the new dividend i.e.
In the divisor put the first digit as double the quotient and place a blank its right. i.e.
Put the largest possible digit to fill the blank which will also become the new digit in the quotient, such that when the new divisor is multiplied to the new quotient the product is less than or equal to the dividend.
As × = 981, we can choose the digit to be filled in the blank as .
Since, the remainder is and there are no more digits left in the given number, we get: 3481 =
We have found the square root of 3481.

(iv) 529

Instructions

Finding the square root using long division method

Starting from right place a bar on top of every pair of digits. If a single digit remains on the left, add a bar to that as well. For eg: 5 29
From the left, find the square which is less than or equal to the digit. Here we have, (22 < 5 < 32). Here, 2 becomes the first digit of the quotient and we get a remainder of when subtracting 22 from 5.
Bring down the number under the next bar (i.e. 29) to the right of the remainder. We now get the new dividend i.e.
In the divisor put the first digit as double the quotient and place a blank its right. i.e. _
Put the largest possible digit to fill the blank which will also become the new digit in the quotient, such that when the new divisor is multiplied to the new quotient the product is less than or equal to the dividend.
As × = 129, we can choose the digit to be filled in the blank as .
Since, the remainder is and there are no more digits left in the given number, we get: 529 =
We have found the square root of 529.

(v) 3249

Instructions

Finding the square root using long division method

Starting from right place a bar on top of every pair of digits. If a single digit remains on the left, add a bar to that as well. For eg: 32 49
From the left, find the square which is less than or equal to the digit. Here we have, (52 < 32 < 62). Here, 5 becomes the first digit of the quotient and we get a remainder of when subtracting 52 from 32.
Bring down the number under the next bar (i.e. 49) to the right of the remainder. We now get the new dividend i.e.
In the divisor put the first digit as double the quotient and place a blank its right. i.e. _
Put the largest possible digit to fill the blank which will also become the new digit in the quotient, such that when the new divisor is multiplied to the new quotient the product is less than or equal to the dividend.
As × = 749, we can choose the digit to be filled in the blank as .
Since, the remainder is and there are no more digits left in the given number, we get: 3249 =
We have found the square root of 3249.

(vi) 1369

Instructions

Finding the square root using long division method

Starting from right place a bar on top of every pair of digits. If a single digit remains on the left, add a bar to that as well. For eg: 13 69
From the left, find the square which is less than or equal to the digit. Here we have, (32 < 13 < 42). Here, 3 becomes the first digit of the quotient and we get a remainder of when subtracting 32 from 13.
Bring down the number under the next bar (i.e. 69) to the right of the remainder. We now get the new dividend i.e.
In the divisor put the first digit as double the quotient and place a blank its right. i.e. _
Put the largest possible digit to fill the blank which will also become the new digit in the quotient, such that when the new divisor is multiplied to the new quotient the product is less than or equal to the dividend.
As × = 469, we can choose the digit to be filled in the blank as .
Since, the remainder is and there are no more digits left in the given number, we get: 1369 =
We have found the square root of 1369.

(vii) 5776

Instructions

Finding the square root using long division method

Starting from right place a bar on top of every pair of digits. If a single digit remains on the left, add a bar to that as well. For eg: 57 76
From the left, find the square which is less than or equal to the digit. Here we have, (72 < 57 < 82). Here, 7 becomes the first digit of the quotient and we get a remainder of when subtracting 72 from 57.
Bring down the number under the next bar (i.e. 76) to the right of the remainder. We now get the new dividend i.e.
In the divisor put the first digit as double the quotient and place a blank its right. i.e. _
Put the largest possible digit to fill the blank which will also become the new digit in the quotient, such that when the new divisor is multiplied to the new quotient the product is less than or equal to the dividend.
As × = 876, we can choose the digit to be filled in the blank as .
Since, the remainder is and there are no more digits left in the given number, we get: 5776 =
We have found the square root of 5776.

(viii) 7921

Instructions

Finding the square root using long division method

Starting from right place a bar on top of every pair of digits. If a single digit remains on the left, add a bar to that as well. For eg: 79 21
From the left, find the square which is less than or equal to the digit. Here we have, (82 < 79 < 92). Here, 8 becomes the first digit of the quotient and we get a remainder of when subtracting 82 from 79.
Bring down the number under the next bar (i.e. 21) to the right of the remainder. We now get the new dividend i.e.
In the divisor put the first digit as double the quotient and place a blank its right. i.e. _
Put the largest possible digit to fill the blank which will also become the new digit in the quotient, such that when the new divisor is multiplied to the new quotient the product is less than or equal to the dividend.
As × = 1521, we can choose the digit to be filled in the blank as .
Since, the remainder is and there are no more digits left in the given number, we get: 7921 =
We have found the square root of 7921.

(ix) 576

Instructions

Finding the square root using long division method

Starting from right place a bar on top of every pair of digits. If a single digit remains on the left, add a bar to that as well. For eg: 5 76
From the left, find the square which is less than or equal to the digit. Here we have, (22 < 5 < 32). Here, 2 becomes the first digit of the quotient and we get a remainder of when subtracting 22 from 5.
Bring down the number under the next bar (i.e. 76) to the right of the remainder. We now get the new dividend i.e. .
In the divisor put the first digit as double the quotient and place a blank its right. i.e. _
Put the largest possible digit to fill the blank which will also become the new digit in the quotient, such that when the new divisor is multiplied to the new quotient the product is less than or equal to the dividend.
As × = 176, we can choose the digit to be filled in the blank as .
Since, the remainder is and there are no more digits left in the given number, we get: 576 =
We have found the square root of 576.

(x) 1024

Instructions

Finding the square root using long division method

Starting from right place a bar on top of every pair of digits. If a single digit remains on the left, add a bar to that as well. For eg: 10 24
From the left, find the square which is less than or equal to the digit. Here we have, (32 < 10 < 42). Here, 3 becomes the first digit of the quotient and we get a remainder of when subtracting 32 from 10.
Bring down the number under the next bar (i.e. 24) to the right of the remainder. We now get the new dividend i.e.
In the divisor put the first digit as double the quotient and place a blank its right. i.e. _
Put the largest possible digit to fill the blank which will also become the new digit in the quotient, such that when the new divisor is multiplied to the new quotient the product is less than or equal to the dividend.
As × = 124, we can choose the digit to be filled in the blank as .
Since, the remainder is and there are no more digits left in the given number, we get: 1024 =
We have found the square root of 1024.

(xi) 3136

Instructions

Finding the square root using long division method

Starting from right place a bar on top of every pair of digits. If a single digit remains on the left, add a bar to that as well. For eg: 31 36
From the left, find the square which is less than or equal to the digit. Here we have, (52 < 31 < 62). Here, 5 becomes the first digit of the quotient and we get a remainder of when subtracting 52 from 31.
Bring down the number under the next bar (i.e. 36) to the right of the remainder. We now get the new dividend i.e.
In the divisor put the first digit as double the quotient and place a blank its right. i.e. _
Put the largest possible digit to fill the blank which will also become the new digit in the quotient, such that when the new divisor is multiplied to the new quotient the product is less than or equal to the dividend.
As × = 636, we can choose the digit to be filled in the blank as .
Since, the remainder is and there are no more digits left in the given number, we get: 3136 =
We have found the square root of 3136.

(xii) 900

Instructions

Finding the square root using long division method

Starting from right place a bar on top of every pair of digits. If a single digit remains on the left, add a bar to that as well. For eg: 9 00
From the left, find the square which is less than or equal to the digit. Here we have, 32 9. Here, 3 becomes the first digit of the quotient and the remainder is .
We have 0 remaining in the dividend. We know that for square-roots: a number containing an even number of zeroes (say m), upon finding its root, gets reduced to a number having zeroes.
Thus, the squareroot of 900 will have zero(s).
There are no more digits left in the given number, we get: 900 =
We have found the square root of 900.

Find the number of digits in the square root of each of the following numbers (without any calculation).

Instructions

We know that if 'n' is number of digits in a square number: Number of digits in the square root = n2 if n is and equal to n+12 if n is .

(i) 64: Here n = i.e. . Thus, number of digits in 64 = 22 =

(ii) 144: Here n = i.e. . Thus, number of digits in 144 = 3+12 =

(iii) 4489: Here n = i.e. . Thus, number of digits in 4489 = 42 =

(iv) 27225: Here n = i.e. . Thus, number of digits in 27225 = 5+12 =

(iv) 390625: Here n = i.e. . Thus, number of digits in 390625 = 62 =

Find the square root of the following decimal numbers.

(i) 2.56

Instructions

Finding the square root using long division method

When finding square-roots of decimal numbers: The procedure remains the same but the overhead line assigning changes. Instead of starting the assignment from the leftmost digit, the assignment starts from the decimal point. Thus, we get: 2 . 56
From the left, find the square which is less than or equal to the digit. Here we have, (12 < 2 < 22). Here, 1 becomes the first digit of the quotient and we get a remainder of when subtracting 12 from 2.
Bring down the number under the next bar (i.e. 56) to the right of the remainder. Since, it comes after a decimal point, we add a decimal after the current quotient as well. We now get the new dividend i.e. .
In the divisor put the first digit as double the quotient and place a blank its right. i.e. _
Put the largest possible digit to fill the blank which will also become the new digit in the quotient, such that when the new divisor is multiplied to the new quotient the product is less than or equal to the dividend.
As × = 156, we can choose the digit to be filled in the blank as .
Since, the remainder is and there are no more digits left in the given number, we get: 2.56 =
We have found the square root of 2.56.

(ii) 7.29

Instructions

Finding the square root using long division method

When finding square-roots of decimal numbers: The procedure remains the same but the overhead line assigning changes. Instead of starting the assignment from the leftmost digit, the assignment starts from the decimal point. Thus, we get: 7 . 29
From the left, find the square which is less than or equal to the digit. Here we have, (22 < 7 < 32). Here, 2 becomes the first digit of the quotient and we get a remainder of when subtracting 22 from 7.
Bring down the number under the next bar (i.e. 29) to the right of the remainder. Since, it comes after a decimal point, we add a decimal after the current quotient as well. We now get the new dividend i.e. .
In the divisor put the first digit as double the quotient and place a blank its right. i.e. _
Put the largest possible digit to fill the blank which will also become the new digit in the quotient, such that when the new divisor is multiplied to the new quotient the product is less than or equal to the dividend.
As × = 329, we can choose the digit to be filled in the blank as .
Since, the remainder is and there are no more digits left in the given number, we get: 7.29 =
We have found the square root of 7.29.

(iii) 51.84

Instructions

Finding the square root using long division method

When finding square-roots of decimal numbers: The procedure remains the same but the overhead line assigning changes. Instead of starting the assignment from the leftmost digit, the assignment starts from the decimal point. Thus, we get: 51 84
From the left, find the square which is less than or equal to the digit. Here we have, (72 < 51 < 82). Here, 7 becomes the first digit of the quotient and we get a remainder of when subtracting 72 from 51.
Bring down the number under the next bar (i.e. 84) to the right of the remainder. Since, it comes after a decimal point, we add a decimal after the current quotient as well. We now get the new dividend i.e. .
In the divisor put the first digit as double the quotient and place a blank its right. i.e. _
Put the largest possible digit to fill the blank which will also become the new digit in the quotient, such that when the new divisor is multiplied to the new quotient the product is less than or equal to the dividend.
As × = 284, we can choose the digit to be filled in the blank as .
Since, the remainder is and there are no more digits left in the given number, we get: 51.84 =
We have found the square root of 51.84.

(iv) 42.25

Instructions

Finding the square root using long division method

When finding square-roots of decimal numbers: The procedure remains the same but the overhead line assigning changes. Instead of starting the assignment from the leftmost digit, the assignment starts from the decimal point. Thus, we get: 42 25
From the left, find the square which is less than or equal to the digit. Here we have, (62 < 42 < 72). Here, 6 becomes the first digit of the quotient and we get a remainder of when subtracting 62 from 42.
Bring down the number under the next bar (i.e. 25) to the right of the remainder. Since, it comes after a decimal point, we add a decimal after the current quotient as well. We now get the new dividend i.e. .
In the divisor put the first digit as double the quotient and place a blank its right. i.e. _
Put the largest possible digit to fill the blank which will also become the new digit in the quotient, such that when the new divisor is multiplied to the new quotient the product is less than or equal to the dividend.
As × = 625, we can choose the digit to be filled in the blank as .
Since, the remainder is and there are no more digits left in the given number, we get: 42.25 =
We have found the square root of 42.25.

(v) 31.36

Instructions

Finding the square root using long division method

When finding square-roots of decimal numbers: The procedure remains the same but the overhead line assigning changes. Instead of starting the assignment from the leftmost digit, the assignment starts from the decimal point. Thus, we get: 31 36
From the left, find the square which is less than or equal to the digit. Here we have, (52 < 31 < 62). Here, 5 becomes the first digit of the quotient and we get a remainder of when subtracting 52 from 31.
Bring down the number under the next bar (i.e. 36) to the right of the remainder. Since, it comes after a decimal point, we add a decimal after the current quotient as well. We now get the new dividend i.e. .
In the divisor put the first digit as double the quotient and place a blank its right. i.e. _
Put the largest possible digit to fill the blank which will also become the new digit in the quotient, such that when the new divisor is multiplied to the new quotient the product is less than or equal to the dividend.
As × = 636, we can choose the digit to be filled in the blank as .
Since, the remainder is and there are no more digits left in the given number, we get: 31.36 =
We have found the square root of 31.36.

Find the least number which must be subtracted from each of the following numbers so as to get a perfect square. Also find the square root of the perfect square so obtained.

Instructions

(i) 402

Using the division method, we see that: 402 gives us a remainder of with the nearest square being (quotient).

Thus, is the least required number to be subtracted from 402 to get a perfect square. That perfect sqrt being 402−2 = 400=

(ii) 1989

Using the division method, we see that: 1989 gives us a remainder of with the nearest square being (quotient).

Thus, is the least required number to be subtracted from 1989 to get a perfect square. That perfect sqrt being 1989−53 = 1936=

(iii) 3250

Using the division method, we see that: 3250 gives us a remainder of with the nearest square being (quotient).

Thus, is the least required number to be subtracted from 3250 to get a perfect square. That perfect sqrt being 3250−1 = 3249=

(iv) 825

Using the division method, we see that: 825 gives us a remainder of with the nearest square being (quotient).

Thus, is the least required number to be subtracted from 825 to get a perfect square. That perfect sqrt being 825−41 = 784=

(v) 4000

Using the division method, we see that: 4000 gives us a remainder of with the nearest square being (quotient).

Thus, is the least required number to be subtracted from 4000 to get a perfect square. That perfect sqrt being 4000−31 = 3969=

Find the least number which must be added to each of the following numbers so as to get a perfect square. Also find the square root of the perfect square so obtained.

Instructions

(i) 525

Using the division method, we see that: 525 gives us a remainder of with the nearest square being (quotient).

Checking the value of 232 = . The smallest number that needs to added to get a perfect square becomes: – =

Thus, the new number is and it is the square of .

(ii) 1750

Using the division method, we see that: 1750 gives us a remainder of with the nearest square being (quotient).

Checking the value of 422 = . The smallest number that needs to added to get a perfect square becomes: – =

Thus, the new number is and it is the square of .

(iii) 252

Using the division method, we see that: 252 gives us a remainder of with the nearest square being (quotient).

Checking the value of 162 = . The smallest number that needs to added to get a perfect square becomes: – =

Thus, the new number is and it is the square of .

(iv) 1825

Using the division method, we see that: 1825 gives us a remainder of with the nearest square being (quotient).

Checking the value of 432 = . The smallest number that needs to added to get a perfect square becomes: – =

Thus, the new number is and it is the square of .

(v) 6412

Using the division method, we see that: 6412 gives us a remainder of with the nearest square being (quotient).

Checking the value of 812 = . The smallest number that needs to added to get a perfect square becomes: – =

Thus, the new number is and it is the square of .

Find the length of the side of a square whose area is 441 m2.

Instructions

Let the length of the side of the square be t.

Thus, area of the square = side2 = t2 m2

Thus, required equation: = 441 ⇒ t = m.

Thus, the length of the side is 21 m.

In a right triangle ABC, ∠B = 90°.

Instructions

(a) If AB = 6 cm, BC = 8 cm, find AC.

Using Pythagoras theorem: AC2=AB2+BC2 ⇒ AC2 = 62+82 = + =

Thus, AC2 = i.e. equal to .

Thus, AC is equal to 10 cm.

(b) If AC = 13 cm, BC = 5 cm, find AB.

Using Pythagoras theorem: AC2=AB2+BC2 ⇒ 132 = AB2+52 ⇒ AB2 = - =

Thus, AB = 144 = cm.

A gardener has 1000 plants. He wants to plant these in such a way that the number of rows and the number of columns remain same. Find the minimum number of plants he needs more for this.

Instructions

Let the number of rows of plants (in the plant bed) be 'x'. Then the number of columns of plants will also be .

The required equation becomes: = ⇒ x =

Upon using the division method, we get a remainder of . Thus, the number of rows equal to the number of columns.

We see that the square of is less than 1000. Thus, we find the square of to find the additional number of plants required.

322 = . The additional plants needed is 1024 - 1000 =

Thus, the minimum number of additional plants required are 24.

There are 500 children in a school. For a P.T. drill they have to stand in such a manner that the number of rows is equal to number of columns. How many children would be left out in this arrangement.

Instructions

Let the number of children in a row be x. Thus, number of columns = .

The required equations: total number of students =

Thus, = 500 ⇒ x =

Upon using the division method, we get a remainder of .

Thus, 16 children will be left out in the drill arrangement.

Squares and Square Roots

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Exercise 5.4

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Squares and Square Roots

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Glossary

Exercise 5.4