# Squares and Square Roots - Class 8 - Mathematics

Renews every month. Cancel anytime

### Your personal doubt-solving assistant

Chatterbot AI gives you 100% accurate answers to your questions in an instant.

## Examples - Squares and Square Roots | R.D. Sharma | Mathematics | Class 8

Is 225 a perfect square? If so, find the number whose square is 225.

Yes, 225 is a perfect square. The number whose square is 225 is **15**.

Here is a visual representation of the number **15** on the number line:

Show that 63504 is a perfect square. Also, find the number whose square is 63504 .

### Improve your grades!

Join English Chatterbox to access detailed and curated answers, and score higher than you ever have in your exams.

Sign up nowShow that 17640 is not a perfect square.

### Improve your grades!

Join English Chatterbox to access detailed and curated answers, and score higher than you ever have in your exams.

Sign up nowFind the smallest number by which 180 must be multiplied so that the product is a perfect square.

### Improve your grades!

Join English Chatterbox to access detailed and curated answers, and score higher than you ever have in your exams.

Sign up nowFind the smallest number by which 25200 should be divided so that the result is a perfect square.

### Improve your grades!

The following numbers are not perfect squares. Give reasons:

(i) 1057

(ii) 23452

(iii) 7928

(iv) 222222

### Improve your grades!

What will be the unit's digit of the squares of the following numbers?

(i) 71

(ii) 599

(iii) 2753

(iv) 1234

### Improve your grades!

Which of the following end with digit 1?

$$ 123^2, 77^2, 82^2, 109^2 $$

### Improve your grades!

Determine whether squares of the following numbers are even or odd.

(i) 213

(ii) 3824

(iii) 9777

(iv) 40028

### Improve your grades!

The following numbers are not perfect squares. Give reason.

(i) 64000

(ii) 89722

(iii) 222000

(iv) 505050

### Improve your grades!

Write a pythagorean triplet whose one member is

(i) 14

(ii) 16

### Improve your grades!

Without adding, find the sum:

(i) $1+3+5+7+9$

(ii) $1+3+5+7+9+11+13+15+17+19$

(iii) $1+3+5+7+9+11+13+15+17+19+21+23$

### Improve your grades!

Express:

(i) 49 as the sum of 7 odd natural numbers.

(ii) 121 as the sum of 11 odd natural numbers.

### Improve your grades!

How many natural numbers lie between squares of the following naturaj numbers?

(i) 12 and 13

(ii) 25 and 26

(iii) 89 and 100

### Improve your grades!

Express each of the following as the sum of two consecutive natural number 8

(i) $21^{2}$

(ii) $13^{2}$

(iii) $19^{2}$

### Improve your grades!

Find whether 55 is a perfect square or not?

### Improve your grades!

Observe the following pattern and find the missing digits:

$$ \begin{aligned} 11^{2} & =121 \\ 101^{2} & =10201 \\ 1001^{2} & =1002001 \\ 10001^{2} & =100020001 \\ 100001^{2} & =1 \ldots 2 \ldots 1 \\ 10000001^{2} & =\ldots \ldots . . \ldots . . . . . \end{aligned} $$

### Improve your grades!

Observe the following pattern and supply the missing numbers:

$$ \begin{aligned} 11^{2} & =121 \\ 101^{2} & =10201 \\ 10101^{2} & =102030201 \\ 1010101^{2} & =\ldots \ldots \ldots \ldots \ldots . . . . . . . . \\ \ldots \ldots . . \ldots . . & =10203040504030201 \end{aligned} $$

### Improve your grades!

Using the given pattern, find the missing numbers:

$$ \begin{gathered} 1^{2}+2^{2}+2^{2}=3^{2} \\ 2^{2}+3^{2}+6^{2}=7^{2} \\ 3^{2}+4^{2}+12^{2}=13^{2} \\ 4^{2}+5^{2}+(\ldots)^{2}=21^{2} \\ 5^{2}+(\ldots)^{2}+30^{2}=31^{2} \\ 6^{2}+7^{2}+(\ldots)^{2}=(\ldots)^{2} \end{gathered} $$

### Improve your grades!

Using suitable patterns, compute the following:

(i) $\frac{333^{2}}{12321}=\ldots$

(ii) $\frac{666666^{2}}{12345654321}=\ldots$

### Improve your grades!

Find the squares of the following numbers using column method:

(i) 25

(ii) 96

### Improve your grades!

Find the squares of the following numbers using column method:

(i) 99

(ii) 89

### Improve your grades!

Find the square of the following numbers by Visual method:

(i) 54

(ii) 97

### Improve your grades!

Find the square of the following numbers by Visual method:

(i) 205

(ii) 315

### Improve your grades!

Find the square of the following numbers using the identity.

$$(a+b)^{2}=a^{2}+2 a b+b^{2}$$

(i) 509

(ii) 211

(iii) 625

### Improve your grades!

Find the square of the following numbers using the identity.

$$ (a-b)^{2}=a^{2}-2 a b+b^{2} $$

(i) 491

(ii) 189

(iii) 575

### Improve your grades!

Find the square root of 11025 by prime factorization.

### Improve your grades!

Find the square root of 7744 by prime factorization.

### Improve your grades!

Find the square root of 298116 by prime factorization.

### Improve your grades!

Find the smallest number by which 1100 must be multiplied square root of the perfect square so obtained.

### Improve your grades!

5929 students are sitting in an auditorium in such a manner that there are as many students in a row as there are rows in the auditorium. How many rows are there in the auditorium?

### Improve your grades!

A general wishing to arrange his men, who were 335250 in number in the form of a square found that there were 9 men left over. How many were there in each row?

### Improve your grades!

The product of two numbers is 1575 and their quotient is $\frac{9}{7}$. Find the numbers.

### Improve your grades!

Find the smallest square number divisible by each one of the numbers 8, 9 and 10 .

### Improve your grades!

Find the square root of each of the following numbers by long division method:

(i) 54756

(ii) 390625

(iii) 4937284

### Improve your grades!

Find the least number which must be subtracted from 18265 to make it a perfect square. Also, find the square root of the resulting number. Explain with steps.

### Improve your grades!

Find the least number which must be added to 306452 to make it a perfect square. Explain with steps.

### Improve your grades!

Find the greatest number of six digits which is a perfect square. Explain with steps.

### Improve your grades!

Find the least number of four digits which is a perfect square. Explain with steps.

### Improve your grades!

## Exercise 3.1 - Squares and Square Roots | R.D. Sharma | Mathematics | Class 8

Which of the following numbers are perfect squares?

(i) 484

(ii) 625

(iii) 576

(iv) 941

(v) 961

(vi) 2500

The numbers that are perfect squares among the given options are:

(i) 484

(ii) 625

(iii) 576

(v) 961

(vi) 2500

**Only (iv) 941 is not a perfect square.**

Show that each of the following numbers is a perfect square. Also, find the number whose square is the given number in each case:

(i) 1156

(ii) 2025

(iii) 14641

(iv) 4761

### Improve your grades!

Find the smallest number by which the given number must be multiplied so that the product is a perfect square:

(i) 23805

(ii) 12150

(iii) 7688

### Improve your grades!

Find the smallest number by which the given number must be divided so that the resulting number is a perfect square:

(i) 14283

(ii) 1800

(iii) 2904

### Improve your grades!

Which of the following numbers are perfect squares?

$$ 11,12,16,32,36,50,64,79,81,111,121 $$

### Improve your grades!

Using prime factorization method, find which of the following numbers are perfect squares?

$$189,225,2048,343,441,2916,11025,3549$$

### Improve your grades!

By what number should each of the following numbers be multiplied to get a perfect square in each case? Also, find the number whose square is the new number.

(i) 8820

(ii) 3675

(iii) 605

(iv) 2880

(v) 4056

(vi) 3468

(vii) 7776

### Improve your grades!

By what numbers should each of the following be divided to get a perfect square it each case? Also, find the number whose square is the new number.

(i) 16562

(ii) 3698

(iii) 5103

(iv) 3174

(v) 1575

### Improve your grades!

Find the greatest number of two digits which is a perfect square.

### Improve your grades!

Find the least number of three digits which is perfect square.

### Improve your grades!

Find the smallest number by which 4851 must be multiplied so that the product becomes a perfect square.

### Improve your grades!

Find the smallest number by which 28812 must be divided so that the quotient becomes a perfect square.

### Improve your grades!

Find the least number by which 1152 must be divided so that it becomes a perfect square. Also, find the number whose square is the resulting number.

### Improve your grades!

## Exercise 3.2 - Squares and Square Roots | R.D. Sharma | Mathematics | Class 8

The following numbers are not perfect squares. Give reason.

(i) 1547

(ii) 45743

(iii) 8948

(iv) 333333

For each of the given numbers, here are the reasons why they are not perfect squares:

**1547**: It is not a perfect square. The nearest perfect square numbers to 1547 are 1521 and 1600.**45743**: It is not a perfect square. The nearest perfect square numbers to 45743 are 45369 and 45796.

**8948**: It is not a perfect square. The nearest perfect square numbers to 8948 are 8836 and 9025.**333333**: It is not a perfect square. The nearest perfect square numbers to 333333 are 332929 and 334084.

The determination of these numbers not being perfect squares is based upon their lack of a whole number square root. Each is situated between two perfect squares, which means no whole number squared equals any of these numbers.

Show that the following numbers are not perfect squares:

(i) 9327

(ii) 4058

(iii) 22453

(iv) 743522

### Improve your grades!

The square of which of the following numbers would be an odd number?

(i) 731

(ii) 3456

(iii) 5559

(iv) 42008

### Improve your grades!

What will be the units digit of the squares of the following numbers?

(i) 52

(ii) 977

(iii) 4583

(iv) 78367

(v) 52698

(vi) 99880

(vii) 12796

(viii) 55555

(ix) 53924

### Improve your grades!

Observe the following pattern

$$ \begin{aligned} 1+3 & =2^{2} \\ 1+3+5 & =3^{2} \\ 1+3+5+7 & =4^{2} \end{aligned} $$

and write the value of $1+3+5+7+9+\cdots$ upto $n$ terms.

### Improve your grades!

Observe the following pattern

$$ \begin{aligned} & 2^{2}-1^{2}=2+1 \\ & 3^{2}-2^{2}=3+2 \\ & 4^{2}-3^{2}=4+3 \\ & 5^{2}-4^{2}=5+4 \end{aligned} $$

and find the value of

(i) $100^{2}-99^{2}$

(ii) $111^{2}-109^{2}$

(iii) $99^{2}-96^{2}$

### Improve your grades!

Which of the following triplets are Pythagorean?

(i) $(8,15,17)$

(ii) $(18,80,82)$

(iii) $(14,48,51)$

(iv) $(10,24,26)$

(v) $(16,63,65)$

(vi) $(12,35,38)$

### Improve your grades!

Observe the following pattern

$$ \begin{aligned} & (1 \times 2)+(2 \times 3)=\frac{2 \times 3 \times 4}{3} \\ & (1 \times 2)+(2 \times 3)+(3 \times 4)=\frac{3 \times 4 \times 5}{3} \\ & (1 \times 2)+(2 \times 3)+(3 \times 4)+(4 \times 5)=\frac{4 \times 5 \times 6}{3} \end{aligned} $$

and find the value of

$$ (1 \times 2)+(2 \times 3)+(3 \times 4)+(4 \times 5)+(5 \times 6) $$

### Improve your grades!

Observe the following pattern

$$ \begin{aligned} 1 & =\frac{1}{2}\{1 \times(1+1)\} \\ 1+2 & =\frac{1}{2}\{2 \times(2+1)\} \\ 1+2+3 & =\frac{1}{2}\{3 \times(3+1)\} \\ 1+2+3+4 & =\frac{1}{2}\{4 \times(4+1)\} \end{aligned} $$

and find the values of each of the following:

(i) $1+2+3+4+5+\ldots+50$

(ii) $31+32+\ldots+50$

### Improve your grades!

Observe the following pattern

$$ \begin{aligned} 1^{2} & =\frac{1}{6}[1 \times(1+1) \times(2 \times 1+1)] \\ 1^{2}+2^{2} & =\frac{1}{6}[2 \times(2+1) \times(2 \times 2+1)] \\ 1^{2}+2^{2}+3^{2} & =\frac{1}{6}[3 \times(3+1) \times(2 \times 3+1)] \\ 1^{2}+2^{2}+3^{2}+4^{2} & =\frac{1}{6}[4 \times(4+1) \times(2 \times 4+1)] \end{aligned} $$

and find the values of each of the following:

(i) $1^{2}+2^{2}+3^{2}+4^{2}+\ldots+10^{2}$

(ii) $5^{2}+6^{2}+7^{2}+8^{2}+9^{2}+10^{2}+11^{2}+12^{2}$

### Improve your grades!

Which of the following numbers are squares of even numbers?

$$121,225,256,324,1296,6561,5476,4489,373758$$

### Improve your grades!

By just examining the units digits, can you tell which of the following cannot be whole squares?

(i) 1026

(ii) 1028

(iii) 1024

(iv) 1022

(v) 1023

(vi) 1027

### Improve your grades!

Write five numbers for which you cannot decide whether they are squares.

### Improve your grades!

Write five numbers which you cannot decide whether they are square just by looking at the unit's digit.

### Improve your grades!

Write true $(\mathrm{T})$ or false $(\mathrm{F})$ for the following statements.

(i) The number of digits in a square number is even.

(ii) The square of a prime number is prime.

(iii) The sum of two square numbers is a square number.

(iv) The difference of two square numbers is a square number.

(v) The product of two square numbers is a square number.

(vi) No square number is negative.

(vii) There is no square number between 50 and 60 .

(viii) There are fourteen square number upto 200.

### Improve your grades!

## Exercise 3.3 - Squares and Square Roots | R.D. Sharma | Mathematics | Class 8

Find the squares of the following numbers using column method. Verify the result $b$, finding the square using the usual multiplication:

(i) 25

(ii) 37

(iii) 54

(iv) 71

(v) 96

Let's apply the column method to find the squares of the given numbers. We will also verify the result using usual multiplication.

### For $(37)^2$:

Here, $a = 3$ and $b = 7$.

Column I: $a^2 = 3^2 = 9$

Column II: $2ab = 2 \times 3 \times 7 = 42$

Column III: $b^2 = 7^2 = 49$

When arranged in the column method format and added, we get:

Carry over 4 from 49 to 42, making Column II = 42 + 4 = 46, and Column III becomes 9.

Carry over 4 from 46 to 9, making Column I = 9 + 4 = 13, and Column II becomes 6.

Result: $13|6|9 = 1369$

Verification by usual multiplication gives 37 * 37.

### For $(54)^2$:

Here, $a = 5$ and $b = 4$.

Column I: $a^2 = 5^2 = 25$

Column II: $2ab = 2 \times 5 \times 4 = 40$

Column III: $b^2 = 4^2 = 16$

Arranged and added:

Carry over 1 from 16 to 40, making Column II = 40 + 1 = 41, and Column III becomes 6.

Carry over 4 from 41 to 25, making Column I = 25 + 4 = 29, and Column II becomes 1.

Result: $29|1|6 = 2916$

Verification by usual multiplication gives 54 * 54.

### For $(71)^2$:

Here, $a = 7$ and $b = 1$.

Column I: $a^2 = 7^2 = 49$

Column II: $2ab = 2 \times 7 \times 1 = 14$

Column III: $b^2 = 1^2 = 1$

Arranged and added:

Carry over is not needed due to small values in Column II and Column III.

The result directly gives $49|14|1$.

Result: $49|14|1$ (Carrying over from Column II: 4 + 49 = 53, and Column II becomes 4) gives $5041$.

Verification by usual multiplication gives 71 * 71.

### For $(96)^2$:

Here, $a = 9$ and $b = 6$.

Column I: $a^2 = 9^2 = 81$

Column II: $2ab = 2 \times 9 \times 6 = 108$

Column III: $b^2 = 6^2 = 36$

Arranged and added:

Carry over 3 from 36 to 108, making Column II = 108 + 3 = 111, and Column III becomes 6.

Carry over 11 from 111 to 81, making Column I = 81 + 11 = 92, and Column II becomes 1.

Result: $92|1|6 = 9216$

Verification with usual multiplication gives 96 * 96.

Let's verify the calculation for $(37)^2$, $(54)^2$, $(71)^2$, and $(96)^2$ using usual multiplication for precision.

### Verification through usual multiplication:

I'll simultaneously calculate $(37)^2$, $(54)^2$, $(71)^2$, and $(96)^2$ to verify our results. The verification results using usual multiplication are as follows:

For $(37)^2$, the result is $1369$.

For $(54)^2$, the result is $2916$.

For $(71)^2$, the result is $5041$.

For $(96)^2$, the result is $9216$.

These results match perfectly with the ones we calculated using the column method.

Find the squares of the following numbers:

(i) 127

(ii) 503

(iii) 451

(iv) 862

(v) 265

### Improve your grades!

Find the squares of the following numbers:

(i) 425

(ii) 575

(iii) 405

(iv) 205

(v) 95

(vi) 745

(vii) 512

(viii) 995

### Improve your grades!

Find the squares of the following numbers using the identity $(a+b)^{2}=a^{2}+2 a b+b^{2}$ :

(i) 405

(ii) 510

(iii) 1001

(iv) 209

(v) 605

### Improve your grades!

Find the squares of the following numbers using the identity $(a-b)^{2}=a^{2}-2 a b+b^{2}$ :

(i) 395

(ii) 995

(iii) 495

(iv) 498

(v) 99

(vi) 999

(vii) 599

### Improve your grades!

Find the squares of the following numbers by visual method:

(i) 52

(ii) 95

(iii) 505

(iv) 702

(v) 99

### Improve your grades!

## Exercise 3.4 - Squares and Square Roots | R.D. Sharma | Mathematics | Class 8

Write the possible unit's digits of the square root of the following numbers. Which of these numbers are odd square roots?

(i) 9801

(ii) 99856

(iii) 998001

(iv) 657666025

Here are the square roots of the given numbers along with their units digits and an indication of whether they are odd or not:

**9801**: The square root is $99$. The unit's digit is $9$, and it is an odd square root.**99856**: The square root is $316$. The unit's digit is $6$, and it is not odd.**998001**: The square root is $999$. The unit's digit is $9$, and it is an odd square root.**657666025**: The square root is $25645$. The unit's digit is $5$, and it is an odd square root.

Hence, the numbers with an odd square root are $9801$, $998001$, and $657666025$.

Find the square root of each of the following by prime factorization.

(i) 441

(ii) 196

(iii) 529

(iv) 1764

(v) 1156

(vi) 4096

(vii) 7056

(viii) 8281

(ix) 11664

(x) 47089

(xi) 24336

(xii) 190969

(xiii) 586756

(xiv) 27225

(xv) 3013696

### Improve your grades!

Find the smallest number by which 180 must be multiplied so that it becomes, perfect square. Also, find the square root of the perfect square so obtained

### Improve your grades!

Find the smallest number by which 147 must be multiplied so that it becomes a perfect square. Also, find the square root of the number so obtained.

### Improve your grades!

Find the smallest number by which 3645 must be divided so that it becomes a perfect square. Also, find the square root of the resulting number.

### Improve your grades!

Find the smallest number by which 1152 must be divided so that it becomes a perfect square. Also, find the square root of the number so obtained.

### Improve your grades!

The product of two numbers is 1296 . If one number is 16 times the other, find the numbers.

### Improve your grades!

A welfare association collected Rs 202500 as donation from the residents. If each paid as many rupees as there were residents, find the number of residents.

### Improve your grades!

A society collected Rs 92.16 . Each member collected as many paise as there were members. How many members were there and how much did each contribute?

### Improve your grades!

A school collected Rs 2304 as fees from its students. If each student paid as many paise as there were students in the school, how many students were there in the school?

### Improve your grades!

The area of a square field is $5184 \mathrm{~m}^{2}$. A rectangular field, whose length is twice its breadth has its perimeter equal to the perimeter of the square field. Find the area the rectangular field.

### Improve your grades!

Find the least square number, exactly divisible by each one of the numbers:

(i) $6,9,15$ and 20

(ii) $8,12,15$ and 20

### Improve your grades!

Find the square roots of 121 and 169 by the method of repeated subtraction.

### Improve your grades!

Write the prime factorization of the following numbers and hence find their square roots.

(i) 7744

(ii) 9604

(iii) 5929

(iv) 7056

### Improve your grades!

The students of class VIII of a school donated Rs 2401 for PM's National Relief Fund Each student donated as many rupees as the number of students in the class. Find the number of students in the class.

### Improve your grades!

A PT teacher wants to arrange maximum possible number of 6000 students in a field that the number of rows is equal to the number of columns. Find the number rows if 71 were left out after arrangement.

### Improve your grades!

## Exercise 3.5 - Squares and Square Roots | R.D. Sharma | Mathematics | Class 8

Find the least number which must be subtracted from the following numbers to make them a perfect square. Explain each with steps.

(i) 2361

(ii) 194491

(iii) 26535

(iv) 16160

(v) 4401624

To find the least number which must be subtracted from each given number to make it a perfect square, we apply the following steps to each case:

**Find the Square Root:**Compute the square root of the given number.**Floor the Result:**Find the largest integer less than or equal to the square root (this is the floor of the square root).**Compute the Perfect Square:**Square the resulted floor value to find the perfect square.**Subtraction:**Subtract the perfect square from the given number to find the required least number.

Let's apply these steps to each given number:

### (i) 2361

Square root: Approximately 48.590

Floor value: 48

Perfect square: $48^2 = 2304$

Required number to subtract: $2361 - 2304 = 57$

### (ii) 194491

Square root: Approximately 441.011

Floor value: 441

Perfect square: $441^2 = 194481$

Required number to subtract: $194491 - 194481 = 10$

### (iii) 26535

Square root: Approximately 162.896

Floor value: 162

Perfect square: $162^2 = 26244$

Required number to subtract: $26535 - 26244 = 291$

### (iv) 16160

Square root: Approximately 127.122

Floor value: 127

Perfect square: $127^2 = 16129$

Required number to subtract: $16160 - 16129 = 31$

### (v) 4401624

Square root: Approximately 2098.005

Floor value: 2098

Perfect square: $2098^2 = 4401604$

Required number to subtract: $4401624 - 4401604 = 20$

These are the least numbers that must be subtracted from each given number to make them a perfect square.

Find the least number which must be added to the following numbers to make them a perfect square. Explain each with steps.

(i) 5607

(ii) 4931

(iii) 4515600

(iv) 37460

(v) 506900

### Improve your grades!

Find the greatest number of 5 digits which is a perfect square.

### Improve your grades!

Find the least number of 4 digits which is a perfect square.

### Improve your grades!

Find the least number of six digits which is a perfect square. Explain with steps.

### Improve your grades!

Find the greatest number of 4 digits which is a perfect square.

### Improve your grades!

A General arranges his soldiers in rows to form a perfect square. He finds that in doing so, 60 soldiers are left out. If the total number of soldiers be 8160 , find the number of soldiers in each row.

### Improve your grades!

The area of a square field is $60025 \mathrm{~m}^{2}$. A man cycles along its boundary at $18 \mathrm{~km} / \mathrm{hr}$. In how much time will he return at the starting point?

### Improve your grades!

The cost of levelling and turfing a square lawn at $\mathrm{Rs} 2.50 \mathrm{per}^{2}$ is $\mathrm{Rs} 13322.50$. Find the cost of fencing it at Rs 5 per metre. Explain with steps.

### Improve your grades!

Find the greatest number of three digits which is a perfect square. Explain with steps.

### Improve your grades!

Find the smallest number which must be added to 2300 so that it becomes a perfect square.

### Improve your grades!

## Exercise 3.6 - Squares and Square Roots | R.D. Sharma | Mathematics | Class 8

Find the square root of:

(i) $\frac{441}{961}$

(ii) $\frac{324}{841}$

(iii) $4 \frac{29}{49}$

(iv) $2 \frac{14}{25}$

(v) $2 \frac{137}{196}$

(vi) $23 \frac{26}{121}$

(vii) $25 \frac{544}{729}$

(viii) $75 \frac{46}{49}$

(ix) $3 \frac{942}{2209}$

(x) $3 \frac{334}{3025}$

(xi) $21 \frac{2797}{3364}$

(xii) $38 \frac{11}{25}$

(xiii) $23 \frac{394}{729}$

(xiv) $21 \frac{51}{169}$

(xv) $10 \frac{151}{225}$

(i) $\sqrt{\frac{441}{961}} = \frac{\sqrt{441}}{\sqrt{961}} = \frac{21}{31}$

(ii) $\sqrt{\frac{324}{841}} = \frac{\sqrt{324}}{\sqrt{841}} = \frac{18}{29}$

(iii) $\sqrt{4\frac{29}{49}} = \sqrt{\frac{225}{49}} = \frac{\sqrt{225}}{\sqrt{49}} = \frac{15}{7}$

(iv) $\sqrt{2\frac{14}{25}} = \sqrt{\frac{64}{25}} = \frac{\sqrt{64}}{\sqrt{25}} = \frac{8}{5}$

(v) $\sqrt{2\frac{14}{25}} = \sqrt{\frac{64}{25}} = \frac{\sqrt{64}}{\sqrt{25}} = \frac{8}{5}$

(vi) $\sqrt{23\frac{26}{121}} = \sqrt{\frac{2809}{121}} = \frac{\sqrt{2809}}{\sqrt{121}} = \frac{53}{11}$

(vii) $\sqrt{25\frac{544}{729}} = \sqrt{\frac{18769}{729}} = \frac{\sqrt{18769}}{\sqrt{729}} = \frac{137}{27}$

(viii) $\sqrt{75 \frac{46}{49}} = \sqrt{\frac{3721}{49}} = \frac{\sqrt{3721}}{\sqrt{49}} = \frac{61}{7}$

(ix) $\sqrt{3 \frac{942}{2209}} = \sqrt{\frac{7569}{2209}} = \frac{\sqrt{7569}}{\sqrt{2209}} = \frac{87}{47}$

(x) $\sqrt{3 \frac{334}{3025}} = \sqrt{\frac{9409}{3025}} = \frac{\sqrt{9409}}{\sqrt{3025}} = \frac{97}{55}$

(xi) $\sqrt{21 \frac{2797}{3364}} = \sqrt{\frac{73441}{3364}} = \frac{\sqrt{73441}}{\sqrt{3364}} = \frac{271}{58}$

(xii) $\sqrt{38 \frac{11}{25}} = \sqrt{\frac{961}{25}} = \frac{\sqrt{961}}{\sqrt{25}} = \frac{31}{5}$

(xiii) $\sqrt{23 \frac{394}{729}} = \sqrt{\frac{17161}{729}} = \frac{\sqrt{17161}}{\sqrt{729}} = \frac{131}{27}$

(xiv) $\sqrt{21 \frac{51}{169}} = \sqrt{\frac{3600}{169}} = \frac{\sqrt{3600}}{\sqrt{169}} = \frac{60}{13}$

(xv) $\sqrt{10 \frac{151}{225}} = \sqrt{\frac{2401}{225}} = \frac{\sqrt{2401}}{\sqrt{225}} = \frac{49}{15}$

Find the value, with intermediate steps:

(i) $\frac{\sqrt{80}}{\sqrt{405}}$

(ii) $\frac{\sqrt{441}}{\sqrt{625}}$

(iii) $\frac{\sqrt{1587}}{\sqrt{1728}}$

(iv) $\sqrt{72} \times \sqrt{338}$

(v) $\sqrt{45} \times \sqrt{20}$

### Improve your grades!

The area of a square field is $80 \frac{244}{729}$ square metres. Find the length of each side of the field. Explain with intermediate steps.

### Improve your grades!

The area of a square field is $30 \frac{1}{4} \mathrm{~m}^{2}$. Calculate the length of the side of the square. Explain with steps.

### Improve your grades!

Find the length of a side of a square playground whose area is equal to the area of a rectangular field of dimensions $72 \mathrm{~m}$ and $338 \mathrm{~m}$. Explain with steps.

### Improve your grades!

## Exercise 3.7 - Squares and Square Roots | R.D. Sharma | Mathematics | Class 8

Find the square root of the following numbers in decimal form:

1. 84.8241

2. 0.7225

3. 0.813604

4. 0.00002025

5. 150.0625

6. 225.6004

7. 3600.720036

8. 236.144689

9. 0.00059049

10. 176.252176

11. 9998.0001

12. 0.00038809

1)

$$\sqrt{84.8241} $$

$$ \sqrt{\frac{848241}{10000}} $$

$$ \frac{\sqrt{3^2×307^2}}{\sqrt{100^2}} $$

$$ \frac{3 \times 307}{100} $$

$$ \frac{921}{100} $$

$$ 9.21 $$

2)

$$\sqrt{0.7225} $$

$$ \sqrt{\frac{7225}{10000}} $$

$$ \frac{\sqrt{5^2×17^2}}{\sqrt{100^2}} $$

$$ \frac{5 \times 17}{100} $$

$$ \frac{85}{100} $$

$$ 0.85 $$

3)

$$\sqrt{0.813604} $$

$$ \sqrt{\frac{813604}{1000000}} $$

$$ \frac{\sqrt{2^2×11^2×41^2}}{\sqrt{1000^2}} $$

$$ \frac{2 \times 11 \times 41}{1000} $$

$$ \frac{902}{1000} $$

$$ 0.902 $$

4)

$$\sqrt{0.00002025} $$

$$ \sqrt{\frac{2025}{100000000}} $$

$$ \frac{\sqrt{3^4×5^2}}{\sqrt{10000^2}} $$

$$ \frac{3^2 \times 5}{10000} $$

$$ \frac{45}{10000} $$

$$ 0.0045 $$

5)

$$\sqrt{150.0625} $$

$$ \sqrt{\frac{1500625}{10000}} $$

$$ \frac{\sqrt{5^4×7^4}}{\sqrt{100^2}} $$

$$ \frac{5^2 \times 7^2}{100} $$

$$ \frac{1225}{100} $$

$$ 12.25 $$

6)

$$\sqrt{225.6004} $$

$$ \sqrt{\frac{2256004}{10000}} $$

$$ \frac{\sqrt{2^2×751^2}}{\sqrt{100^2}} $$

$$ \frac{2 \times 751}{100} $$

$$ \frac{1502}{100} $$

$$ 15.02 $$

7)

$$\sqrt{3600.720036} $$

$$ \sqrt{\frac{3600720036}{1000000}} $$

$$ \frac{\sqrt{2^2×3^2×73^2×137^2}}{\sqrt{1000^2}} $$

$$ \frac{2 \times 3 \times 73 \times 137}{1000} $$

$$ \frac{60006}{1000} $$

$$ 60.006 $$

8)

$$\sqrt{236.144689} $$

$$ \sqrt{\frac{236144689}{1000000}} $$

$$ \frac{\sqrt{11^4×127^2}}{\sqrt{1000^2}} $$

$$ \frac{11^2 \times 127}{1000} $$

$$ \frac{15367}{1000} $$

$$ 15.367 $$

9)

$$\sqrt{0.00059049} $$

$$ \sqrt{\frac{59049}{100000000}} $$

$$ \frac{\sqrt{3^10}}{\sqrt{10000^2}} $$

$$ \frac{3^5}{10000} $$

$$ \frac{243}{10000} $$

$$ 0.0243 $$

10)

$$\sqrt{176.252176} $$

$$ \sqrt{\frac{176252176}{1000000}} $$

$$ \frac{\sqrt{2^4×3319^2}}{\sqrt{1000^2}} $$

$$ \frac{2^2\times3319}{1000} $$

$$ \frac{13276}{1000} $$

$$ 13.276 $$

10)

$$\sqrt{9998.0001} $$

$$ \sqrt{\frac{99980001}{10000}} $$

$$ \frac{\sqrt{3^4×11^2×101^2}}{\sqrt{100^2}} $$

$$ \frac{3^2 \times 11 \times 101}{100} $$

$$ \frac{9999}{100} $$

$$ 99.99 $$

11)

$$ \sqrt{0.00038809} $$

$$ \sqrt{\frac{38809}{100000000}} $$

$$ \frac{\sqrt{197^2}}{\sqrt{10000^2}} $$

$$ \frac{197}{10000} $$

$$ 0.0197 $$

What is that fraction which when multiplied by itself gives 227.798649 ? Explain with steps

### Improve your grades!

The area of a square playground is 256.6404 square metres. Find the length of one side of the playground. Explain with steps.

### Improve your grades!

What is the fraction which when multiplied by itself gives 0.00053361 ? Explain with steps.

### Improve your grades!

Simplify:

(i) $\frac{\sqrt{59.29}-\sqrt{5.29}}{\sqrt{59.29}+\sqrt{5.29}}$

(ii) $\frac{\sqrt{0.2304}+\sqrt{0.1764}}{\sqrt{0.2304}-\sqrt{0.1764}}$

Explain with steps.

### Improve your grades!

Evaluate $\sqrt{50625}$ and hence find the value of $\sqrt{506.25}+\sqrt{5.0625}$

Explain with steps.

### Improve your grades!

Find the value of $\sqrt{103.0225}$ and hence find the value of

(i) $\sqrt{10302.25}$

(ii) $\sqrt{1.030225}$

### Improve your grades!

## Exercise 3.8 - Squares and Square Roots | R.D. Sharma | Mathematics | Class 8

Find the square root of each of the following correct to three places of decimal.

(i) 5

(ii) 7

(iii) 17

(iv) 20

(v) 66

(vi) 427

(vii) 1.7

(viii) 23.1

(ix) 2.5

(x) 237.615

(xi) 15.3215

(xii) 0.9

(xiii) 0.1

(xiv) 0.016

(xv) 0.00064

(xvi) 0.019

(xvii) $\frac{7}{8}$

(xviii) $\frac{5}{12}$

(xix) $2 \frac{1}{2}$

(xx) $287 \frac{5}{8}$

Here are the square roots of the given numbers correct to three decimal places:

$$\sqrt{5} = 2.236$$

$$\sqrt{7} = 2.646$$

$$\sqrt{17} = 4.123$$

$$\sqrt{20} = 4.472$$

$$\sqrt{66} = 8.124$$

$$\sqrt{427} = 20.664$$

$$\sqrt{1.7} = 1.304$$

$$\sqrt{23.1} = 4.806$$

$$\sqrt{2.5} = 1.581$$

$$\sqrt{237.615} = 15.415$$

$$\sqrt{15.3215} = 3.914$$

$$\sqrt{0.9} = 0.949$$

$$\sqrt{0.1} = 0.316$$

$$\sqrt{0.016} = 0.126$$

$$\sqrt{0.00064} = 0.025$$

$$\sqrt{0.019} = 0.138$$

$$\sqrt{\frac{7}{8}} = 0.935$$

$$\sqrt{\frac{5}{12}} = 0.645$$

$$\sqrt{2 \frac{1}{2}} = 1.581$$

$$\sqrt{287 \frac{5}{8}} = 16.960$$

These values provide the required square roots to three decimal places.

Find the square root of 12.0068 correct to four decimal places using long division.

### Improve your grades!

Find the square root of 11 correct to five decimal places using long division.

### Improve your grades!

Given that: $\sqrt{2}=1.414, \sqrt{3}=1.732, \sqrt{5}=2.236$ and $\sqrt{7}=2.646$, evaluate each of the following:

(i) $\sqrt{\frac{144}{7}}$

(ii) $\sqrt{\frac{2500}{3}}$

### Improve your grades!

Given that $\sqrt{2}=1.414, \sqrt{3}=1.732, \sqrt{5}=2.236$ and $\sqrt{7}=2.646$, find the square roots of the following:

(i) $\frac{196}{75}$

(ii) $\frac{400}{63}$

(iii) $\frac{150}{7}$

(iv) $\frac{256}{5}$

(v) $\frac{27}{50}$

### Improve your grades!

## Exercise 3.9 - Squares and Square Roots | R.D. Sharma | Mathematics | Class 8

Using square root table, find the square roots of the following:

1. 7

2. 15

3. 74

4. 82

5. 198

6. 540

7. 8700

8. 3509

9. 6929

10. 25725

11. 1312

12. 4192

13. 4955

14. $\frac{99}{144}$

15. $\frac{57}{169}$

16. $\frac{101}{169}$

17. 13.21

18. 21.97

19. 110

20. 1110

21. 11.11

Here are the square roots of the specified numbers, calculated using a square root table and rounded to relevant decimal places:

$$\sqrt{7} = 2.646$$

$$\sqrt{15} = 3.873$$

$$\sqrt{74} = 8.602$$

$$\sqrt{82} = 9.055$$

$$\sqrt{198} = 14.071$$

$$\sqrt{540} = 23.238$$

$$\sqrt{8700} = 93.274$$

$$\sqrt{3509} = 59.237$$

$$\sqrt{6929} = 83.241$$

$$\sqrt{25725} = 160.39$$

$$\sqrt{1312} = 36.222$$

$$\sqrt{4192} = 64.746$$

$$\sqrt{4955} = 70.392$$

$$\sqrt{\frac{99}{144}} = 0.829$$

$$\sqrt{\frac{57}{169}} = 0.581$$

$$\sqrt{\frac{101}{169}} = 0.773$$

$$\sqrt{13.21} = 3.634$$

$$\sqrt{21.97} = 4.687$$

$$\sqrt{110} = 10.488$$

$$\sqrt{1110} = 33.317$$

$$\sqrt{11.11} = 3.333$$

These values represent the square roots with the precision available from the square root table or computational methods used.

The area of a square field is $325 \mathrm{~m}^{2}$. Find the approximate length of one side of the field. Explain with steps.

### Improve your grades!

Find the length of a side of a square, whose area is equal to the area of a rectangle with sides $240 \mathrm{~m}$ and $70 \mathrm{~m}$. Explain with steps.