8类NCERT解决方案-第6章平方根和平方根–练习6.4

问题1.通过除法找到以下每个数字的平方根。

(i)2304

解决方案：

Step 1: Place a bar over every pair of digits starting from the digit at one’s place. If the number of digits in it is odd, then the left-most single digit too will have a bar. 

Thus we have, $\overline{23}$ $\overline{04}$

Step 2: Find the largest number whose square is less than or equal to the number under the extreme left bar. Take this number as the divisor and the quotient with the number under the extreme left bar as the dividend. 

Here, we have 23

Divide and get the remainder.

Here, we get 7

Step 3: Bring down the remaining number under the next bar to the right of the remainder. 

Here, its 04

So now the new dividend is 704.

Step 4: Double the quotient and enter it with a blank on its right. 

Step 5: Guess a 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.

In this case 88 × 8 = 704.

So we choose the new digit as 8. 

Get the remainder.

Step 6: Since the remainder is 0 and no digits are left in the given number. 

Hence, √2304 = 48

(ii)4489

解决方案：

Step 1: Place a bar over every pair of digits starting from the digit at one’s place. If the number of digits in it is odd, then the left-most single digit too will have a bar.

Thus we have, $\overline{44}$ $\overline{89}$

Step 2: Find the largest number whose square is less than or equal to the number under the extreme left bar. Take this number as the divisor and the quotient with the number under the extreme left bar as the dividend.

Here, we have 44

Divide and get the remainder.

Here, we get 8

Step 3: Bring down the remaining number under the next bar to the right of the remainder.

Here, its 89

So now the new dividend is 889.

Step 4: Double the quotient and enter it with a blank on its right.

Step 5: Guess a 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.

In this case 127 × 7 = 889.

So we choose the new digit as 7.

Get the remainder.

Step 6: Since the remainder is 0 and no digits are left in the given number.

Hence, √4489 = 67

(iii)3481

解决方案：

Step 1: Place a bar over every pair of digits starting from the digit at one’s place. If the number of digits in it is odd, then the left-most single digit too will have a bar.

Thus we have, $\overline{34}$ $\overline{81}$

Step 2: Find the largest number whose square is less than or equal to the number under the extreme left bar. Take this number as the divisor and the quotient with the number under the extreme left bar as the dividend.

Here, we have 34

Divide and get the remainder.

Here, we get 9

Step 3: Bring down the remaining number under the next bar to the right of the remainder.

Here, its 81

So now the new dividend is 981.

Step 4: Double the quotient and enter it with a blank on its right.

Step 5: Guess a 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.

In this case 109 × 9 = 981.

So we choose the new digit as 9.

Get the remainder.

Step 6: Since the remainder is 0 and no digits are left in the given number.

Hence, √3481 = 59

(iv)529

解决方案：

Step 1: Place a bar over every pair of digits starting from the digit at one’s place. If the number of digits in it is odd, then the left-most single digit too will have a bar.

Thus we have, $\overline{5}$ $\overline{29}$

Step 2: Find the largest number whose square is less than or equal to the number under the extreme left bar. Take this number as the divisor and the quotient with the number under the extreme left bar as the dividend.

Here, we have 5

Divide and get the remainder.

Here, we get 1

Step 3: Bring down the remaining number under the next bar to the right of the remainder.

Here, its 29

So now the new dividend is 129.

Step 4: Double the quotient and enter it with a blank on its right.

Step 5: Guess a 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.

In this case 23 × 3 = 129.

So we choose the new digit as 3.

Get the remainder.

Step 6: Since the remainder is 0 and no digits are left in the given number.

Hence, √529 = 23

(v)3249

解决方案：

Step 1: Place a bar over every pair of digits starting from the digit at one’s place. If the number of digits in it is odd, then the left-most single digit too will have a bar.

Thus we have, $\overline{32}$ $\overline{49}$

Step 2: Find the largest number whose square is less than or equal to the number under the extreme left bar. Take this number as the divisor and the quotient with the number under the extreme left bar as the dividend.

Here, we have 32

Divide and get the remainder.

Here, we get 7

Step 3: Bring down the remaining number under the next bar to the right of the remainder.

Here, its 49

So now the new dividend is 749.

Step 4: Double the quotient and enter it with a blank on its right.

Step 5: Guess a 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.

In this case 107 × 7 = 749.

So we choose the new digit as 7.

Get the remainder.

Step 6: Since the remainder is 0 and no digits are left in the given number.

Hence, √3249 = 57

(vi)1369

解决方案：

Step 1: Place a bar over every pair of digits starting from the digit at one’s place. If the number of digits in it is odd, then the left-most single digit too will have a bar.

Thus we have, $\overline{13}$ $\overline{69}$

Step 2: Find the largest number whose square is less than or equal to the number under the extreme left bar. Take this number as the divisor and the quotient with the number under the extreme left bar as the dividend.

Here, we have 13

Divide and get the remainder.

Here, we get 4

Step 3: Bring down the remaining number under the next bar to the right of the remainder.

Here, its 69

So now the new dividend is 469.

Step 4: Double the quotient and enter it with a blank on its right.

Step 5: Guess a 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.

In this case 37 × 7 = 469.

So we choose the new digit as 7.

Get the remainder.

Step 6: Since the remainder is 0 and no digits are left in the given number.

Hence, √1369 = 37

(vii)5776

解决方案：

Step 1: Place a bar over every pair of digits starting from the digit at one’s place. If the number of digits in it is odd, then the left-most single digit too will have a bar.

Thus we have, $\overline{57}$ $\overline{76}$

Step 2: Find the largest number whose square is less than or equal to the number under the extreme left bar. Take this number as the divisor and the quotient with the number under the extreme left bar as the dividend.

Here, we have 57

Divide and get the remainder.

Here, we get 8

Step 3: Bring down the remaining number under the next bar to the right of the remainder.

Here, its 76

So now the new dividend is 876.

Step 4: Double the quotient and enter it with a blank on its right.

Step 5: Guess a 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.

In this case 146 × 6 = 876.

So we choose the new digit as 6.

Get the remainder.

Step 6: Since the remainder is 0 and no digits are left in the given number.

Hence, √5776 = 76

(viii)7921

解决方案：

Step 1: Place a bar over every pair of digits starting from the digit at one’s place. If the number of digits in it is odd, then the left-most single digit too will have a bar.

Thus we have, $\overline{79}$ $\overline{21}$

Step 2: Find the largest number whose square is less than or equal to the number under the extreme left bar. Take this number as the divisor and the quotient with the number under the extreme left bar as the dividend.

Here, we have 79

Divide and get the remainder.

Here, we get 15

Step 3: Bring down the remaining number under the next bar to the right of the remainder.

Here, its 21

So now the new dividend is 1521.

Step 4: Double the quotient and enter it with a blank on its right.

Step 5: Guess a 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.

In this case 169 × 9 = 1521.

So we choose the new digit as 9.

Get the remainder.

Step 6: Since the remainder is 0 and no digits are left in the given number.

Hence, √7921 = 89

(ix)576

解决方案：

Step 1: Place a bar over every pair of digits starting from the digit at one’s place. If the number of digits in it is odd, then the left-most single digit too will have a bar.

Thus we have, $\overline{5}$ $\overline{76}$

Step 2: Find the largest number whose square is less than or equal to the number under the extreme left bar. Take this number as the divisor and the quotient with the number under the extreme left bar as the dividend.

Here, we have 5

Divide and get the remainder.

Here, we get 1

Step 3: Bring down the remaining number under the next bar to the right of the remainder.

Here, its 76

So now the new dividend is 176.

Step 4: Double the quotient and enter it with a blank on its right.

Step 5: Guess a 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.

In this case 44 × 4 = 176.

So we choose the new digit as 4.

Get the remainder.

Step 6: Since the remainder is 0 and no digits are left in the given number.

Hence, √576 = 24

[x)1024

解决方案：

Step 1: Place a bar over every pair of digits starting from the digit at one’s place. If the number of digits in it is odd, then the left-most single digit too will have a bar.

Thus we have, $\overline{10}$ $\overline{24}$

Step 2: Find the largest number whose square is less than or equal to the number under the extreme left bar. Take this number as the divisor and the quotient with the number under the extreme left bar as the dividend.

Here, we have 10

Divide and get the remainder.

Here, we get 1

Step 3: Bring down the remaining number under the next bar to the right of the remainder.

Here, its 24

So now the new dividend is 124.

Step 4: Double the quotient and enter it with a blank on its right.

Step 5: Guess a 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.

In this case 62 × 2 = 124.

So we choose the new digit as 2.

Get the remainder.

Step 6: Since the remainder is 0 and no digits are left in the given number.

Hence, √1024 = 32

(xi)3136

解决方案：

Step 1: Place a bar over every pair of digits starting from the digit at one’s place. If the number of digits in it is odd, then the left-most single digit too will have a bar.

Thus we have, $\overline{31}$ $\overline{36}$

Step 2: Find the largest number whose square is less than or equal to the number under the extreme left bar. Take this number as the divisor and the quotient with the number under the extreme left bar as the dividend.

Here, we have 31

Divide and get the remainder.

Here, we get 6

Step 3: Bring down the remaining number under the next bar to the right of the remainder.

Here, its 36

So now the new dividend is 636.

Step 4: Double the quotient and enter it with a blank on its right.

Step 5: Guess a 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.

In this case 106 × 6 = 636.

So we choose the new digit as 6.

Get the remainder.

Step 6: Since the remainder is 0 and no digits are left in the given number.

Hence, √3136 = 56

(xii)900

解决方案：

Step 1: Place a bar over every pair of digits starting from the digit at one’s place. If the number of digits in it is odd, then the left-most single digit too will have a bar.

Thus we have, $\overline{9}$ $\overline{00}$

Step 2: Find the largest number whose square is less than or equal to the number under the extreme left bar. Take this number as the divisor and the quotient with the number under the extreme left bar as the dividend.

Here, we have 9

Divide and get the remainder.

Here, we get 0

Step 3: Bring down the remaining number under the next bar to the right of the remainder.

Here, its 0

So now the new dividend is 000.

Step 4: Double the quotient and enter it with a blank on its right.

Step 5: Guess a 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.

In this case 60 × 0 = 000.

So we choose the new digit as 0.

Get the remainder.

Step 6: Since the remainder is 0 and no digits are left in the given number.

Hence, √900 = 30

问题2.在以下每个数字的平方根中找到位数(无需任何计算)。

If n is number of digits in a square number then

Number of digits in the square root = $\frac {n}{2}$if n is even 

and, $\frac{n+1}{2}$ if n is odd.

(i)64

解决方案：

Here, n = 2, which is even

So, number of digits in square root is = $\frac {n}{2}$

= $\frac {2}{2}$

= 1

(ii)144

解决方案：

Here, n = 3, which is odd

So, number of digits in square root is = $\frac {n+1}{2}$

= $\frac {4}{2}$

= 2

(iii)4489

解决方案：

Here, n = 4, which is even

So, number of digits in square root is =$\frac {n}{2}$

= $\frac {4}{2}$

= 2

(iv)27225

解决方案：

Here, n = 5, which is odd

So, number of digits in square root is = $\frac {n+1}{2}$

= $\frac {6}{2}$

= 3

(v)390625

解决方案：

Here, n = 6, which is even

So, number of digits in square root is = $\frac {n}{2}$

= $\frac {6}{2}$

= 3

问题3.找到以下十进制数字的平方根。

(i)2.56

解决方案：

To find the square root of a decimal number we put bars on the integral part of the number in the usual manner. And place bars on the decimal part on every pair of digits beginning with the first decimal place. 

We get $\overline{2}.\overline{56}$

Since the remainder is 0 and no digits are left in the given number.

Hence, √2.56 = 1.6

(ii)7.29

解决方案：

To find the square root of a decimal number we put bars on the integral part of the number in the usual manner. And place bars on the decimal part on every pair of digits beginning with the first decimal place.

We get $\overline{7}.\overline{29}$

Since the remainder is 0 and no digits are left in the given number.

Hence, √7.29 = 2.7

(iii)51.84

解决方案：

To find the square root of a decimal number we put bars on the integral part of the number in the usual manner. And place bars on the decimal part on every pair of digits beginning with the first decimal place.

We get $\overline{51}.\overline{84}$

Since the remainder is 0 and no digits are left in the given number.

Hence, √51.84 = 7.2

(iv)42.25

解决方案：

To find the square root of a decimal number we put bars on the integral part of the number in the usual manner. And place bars on the decimal part on every pair of digits beginning with the first decimal place.

We get $\overline{42}.\overline{25}$

Since the remainder is 0 and no digits are left in the given number.

Hence, √42.25 = 6.5

(v)31.36

解决方案：

To find the square root of a decimal number we put bars on the integral part of the number in the usual manner. And place bars on the decimal part on every pair of digits beginning with the first decimal place.

We get $\overline{31}.\overline{36}$

Since the remainder is 0 and no digits are left in the given number.

Hence, √31.36 = 5.6

问题4.找到以下每个数字必须减去的最小数字，以得到一个完美的平方。还要找到如此获得的理想平方的平方根。

Here, remainder is the least required number to be subtracted from the given number to get a perfect square.

(i)402

解决方案：

By following all the steps for obtaining square root, we get

Here remainder is 2

2 is the least required number to be subtracted from 402 to get a perfect square

New number = 402 – 2 = 400

Thus, √400 = 20

(ii)1989年

解决方案：

By following all the steps for obtaining square root, we get

Here remainder is 53

53 is the least required number to be subtracted from 1989.

New number = 1989 – 53 = 1936

Thus, √1936 = 44

(iii)3250

解决方案：

By following all the steps for obtaining square root, we get

Here remainder is 1

1 is the least required number to be subtracted from 3250 to get a perfect square.

New number = 3250 – 1 = 3249

Thus, √3249 = 57

(iv)825

解决方案：

By following all the steps for obtaining square root, we get

Here, the remainder is 41

41 is the least required number which can be subtracted from 825 to get a perfect square.

New number = 825 – 41 = 784

Thus, √784 = 28

(v)4000

解决方案：

By following all the steps for obtaining square root, we get

Here, the remainder is 31

31 is the least required number which should be subtracted from 4000 to get a perfect square.

New number = 4000 – 31 = 3969

Thus, √3969 = 63

问题5.找到必须加到以下每个数字中的最小数字，以得到一个完美的平方。还要找到如此获得的理想平方的平方根。

(i)525

解决方案：

By following all the steps for obtaining square root, we get

Here remainder is 41

It represents that 22² is less than 525.

Next number is 23, 

Where, 23² = 529

Hence, the number to be added = 529 – 525 = 4

New number = 525+4 = 529

Thus, √529 = 23

(ii)1750年

解决方案：

By following all the steps for obtaining square root, we get

Here the remainder is 69

It represents that 41² is less than in 1750.

The next number is 42 

Where, 42² = 1764

Hence, number to be added to 1750 = 1764 – 1750 = 14

New number = 1750 + 14 = 1764

√1764 = 42

(iii)252

解决方案：

By following all the steps for obtaining square root, we get

Here the remainder is 27.

It represents that 15² is less than 252.

The next number is 16 

Where,16² = 256

Hence, number to be added to 252 = 256 – 252 = 4

New number = 252 + 4 = 256

and √256 = 16

(iv)1825年

解决方案：

By following all the steps for obtaining square root, we get

The remainder is 61.

It represents that 42² is less than in 1825.

Next number is 43 

Where, 43² = 1849

Hence, number to be added to 1825 = 1849 – 1825 = 24

New number = 1825 + 24 = 1849

and √1849 = 43

(v)6412

解决方案：

By following all the steps for obtaining square root, we get

Here, the remainder is 12.

It represents that 80² is less than in 6412.

The next number is 81 

Where, 81² = 6561

Hence, the number to be added = 6561 – 6412 = 149

New number = 6412 + 149 = 6561

and √6561 = 81

问题6.求出面积为441 m ²的正方形边长。

解决方案：

Let the side of square be x m.

Area of square = x²

 According to the given question,

x² = 441

x = √441

Hence, the side of square is 21 m.

问题7.在直角三角形ABC中，∠B= 90°。

(a)如果AB = 6厘米，BC = 8厘米，找到AC

解决方案：

In right triangle ABC

AC² = AB² + BC²  [By Pythagoras Theorem]

AC² = 6² + 8²

AC² = 100

AC = √100

AC = 10 cm

(b)如果AC = 13厘米，BC = 5厘米，找到AB

解决方案：

In right triangle ABC

AC² = AB² + BC²  [By Pythagoras Theorem]

13² = AB² + 5²

AB² = 13² – 5²

AB² = (13+5) (13-5)

AB² = 18 × 8

AB² = 144

AB = √144

AB = 12 cm

问题8.园丁有1000株植物。他想以行数和列数保持相同的方式来植入它们。找到他为此需要更多的最小植物数量。

解决方案：

Let the number of rows and columns be x.

Total number of plants = x²

x² = 1000 

x = √1000

Here the remainder is 39

So the 31² is less than 1000.

Next number is 32 

Where, 32² = 1024

Hence the number to be added = 1024 – 1000 = 24

Hence, the minimum number of plants required by him = 24.

问题9.一所学校有500个孩子。对于PT钻，它们必须以行数等于列数的方式站立。在这种安排中将有多少个孩子被排除在外。

解决方案：

Let the number of rows and columns be x.

Total number of plants = x²

x² = 500

x = √500

Here the remainder is 16

New Number 500 – 16 = 484

and, √484 = 22

Thus, 16 students will be left out in this arrangement.