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Class: Subject: Topic: VIII Mathematics Squares and Square Roots

The list of Subtopics Why these? Why should I care? Common Mistakes

Squares of numbers

Square rootsKnowing where to usesquares and square roots

Properties of squarenumbers

Patterns of a square of anumber

Finding the square of anumber

Finding square roots

Square roots of a decimal

Estimating square roots

Unable to differentiatebetween problems that askfor squares versus those thatask for square roots

Placing bars over numbersfor finding square rootthrough long divisionmethod

Application of squares andsquare roots is used in awide range of topics frommensuration and trigonom-etry to calculus

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1. Introduction

You know that the area of a square =

Let us observe the following table:

Side of a square (in

cm)

Area of the square (in cm2)

4 4 × 4 = 16 = 42 7 7 × 7 = 49 = 72 9 9 × 9 = 81 = 92

11 11 × 11 = 121 = 112 20 20 × 20 = 400 = 202 x x × x = x2 p p × p = p2

What is special about the numbers 4, 9, 25, 64 and other such numbers?

Since 4 can be expressed as 2 × 2 = 22, 9 can be expressed as 3 × 3 = 32,all such numbers can be expressed as the product of the number with itself.

Such numbers like 1, 4, 9, 16, 25, ..... are known as square numbers of 1, 2, 3, 4, 5,......respectively.

Squares of numbers

1. The square of a number is the product of .

Example:

The square of 5 is 5 × 5 = 25.

The square of 13 is

13 ×

13 =

19 .

The square of – 0.4 is (–0.4) × (–0.4) = 0.16.

2. The square of a number is written using the square notation.

Eg: (–2) × (–2) is written as .

7 × 7 is written as .18 ×

18 is written as .

3. 62 is read as ‘6 squared’ or ‘the square of 6’ or ‘6 to the power of 2’.

4. The square notation is expanded to obtain the product of the squared number.

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Perfect square

A natural number is said to be a perfect square, if it is the square of some other naturalnumber.

Example: 81 = 92, 4 = 22, 36 = 62, 10000 = 1002.

Identifying a perfect square:

A given number is perfect square, if it can be expressed as the product of a pair of equalfactors.

Note: The factors can be found by prime factorisation method.

Example1: Is 225 a perfect square ?

Resolving 225 into prime factors,

5 22545

931

533

we get, 225 = 5 × 5 × 3 × 3

As it is expressed as the product of equal factors hence, it is a perfect square.

Example2: Is 180 a perfect square ?

Resolving 180 into prime factors, we get, 180 = 2 × 2 × 3 × 3 × 5

Though 2 and 3 are paired, 5 is not paired.

Thus, 180 cannot be expressed as the product of pairs of equal factors.

Hence, 180 is not a perfect square.

Think: If you were told to multiply 180 by the smallest natural number to make it aperfect square, what would that number be?

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Some more examples:

Perfect squares

Non-perfect squares

9 = 3 × 3 = 32 5 = ? 25 = 5 × 5 = 52 21 = ?

Note: Though 5 can be expressed as 5 5 it is not a prefect square because 5 is nota natural number.

Similarly, 21 is also not a perfect square.

Where to square and where to find the square root?

E.g. 1 The area of a circle is 314 cm2. Find its radius.

We k n ow t h at ar ea of a ci r cle is given by t h e for m u la,

Area = 314 cm2 =

=

314

Thus, r =

Tick the correct answer: In the example shown here, we calculated the:

a) Square

b) Square root

c) None of the above

E.g. 2 The side of a chess board is 12 cm. Find its area.

Here, we need to calculate the square.

Area =

=

=

E.g. 3 The side of a square is 4 cm. What is the length of its diagonal?

The square of the length of the diagonal = sum of squares of the lengths of the sides.(This is an application of the theorem).

i.e. Diagonal2 = (Side1)2 + (Side2)2

Diagonal2 = 42 + 42 = 16 + 16 = 32 cm2

Thus, Diagonal = 32 cm

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Square roots

Square roots of numbers

1. The square root of a positive number is a number which, when multiplied by itself,equals the given positive number.

Example:

What is the square root of 25?

The square root of 25 is 5 because when 5 is multiplied by itself, the product equals25.

5 × 5 = 25

The square root of 25 is 5.

2. The symbol for square root is .

Example:

The square root of 25 is denoted as 25 .

25 = 5

25 is read as ‘the square root of twenty five’.

Properties of squares of numbers

i) The square of an even number is always an even number.

Example:

6 is an even number and 62 = 36 which is even.

8 is an even number and 82 = 64 which is even.

ii) The square of an odd number is always an odd number.

Example:

7 is an odd number and 72 = 49 which is odd.

11 is an odd number and 112 = 121 which is odd.

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Patterns of square numbers

i) Sum of first n odd natural numbers is n2.

Example:

1 = 1 = 12

[one odd number]

1 + 3 = 4 = 22

[sum of first two odd numbers]

1 + 3 + 5 = 9 = 32

[sum of first three odd numbers]

1 + 3 + 5 + 7 [ ] = 16 = 42

1 + 3 + 5 + 7 + 9 [ ] = 25 = 52

1 + 3 + 5 + 7 + 9 + 11 [ ] = 36 = 62

ii) Observe the squares of numbers 1, 11, 111...etc. They give a beautiful pattern:

12 =

112 =

1112 =

11112 =

111112 =

1111112 =

11111112 =

iii) Another interesting pattern:

72 = 49

672 = 4489

6672 = 444889

66672 = 44448889

666672 = 4444488889

6666672 = 444444888889

1

1 2 1

1 2 3 2 1

1 2 3 4 2 3 1

1 2 3 4 5 4 3 2 1

1 2 3 4 5 6 5 4 3 2 1

1 2 3 4 5 6 7 6 5 4 3 2 1

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The numbers 4489, 444889, . . . are obtained by inserting 48 into the middle of thepreceding squares of integers 7, 67, 667, 6667, . . . respectively.

iv) The difference of squares of successive numbers is equal to their sum.

Example: 172 – 162 = 17 + 16

1012 – 1002 = 101 + 100

2352 – 2342 = 235 + 234

In general, (n + 1)2 - ( n )2 = (n + 1+ n)

= (n+1) + (n)

1 1

1

n n x

n n

v) If 1 is added to the product of two consecutive even natural numbers, it is equal to thesquare of the only odd natural number between them:

2 × 4 + 1 = 9 = 32

4 × 6 + 1 = 52

6 × 8 + 1 =

8 × 10 + 1 =

10 × 12 + 1 =

vi) The squares of natural numbers like 11, 111, ….., etc. have a nice pattern as shown below

121 × (1 + 2 + 1) = 484 = 222

112 × (sum of the digits in 112) = (2 ×11)2

12321 × (1 + 2 + 3 + 2 + 1) = 110889 = 3332

1112 × (sum of the digits in 1112) = (3× 111)2

1234321 × (1 + 2 + 3 + 4 + 3 + 2 + 1)

= 19749136 = (4444)2

11112 × (sum of the digits in 11112) = (4 × 1111)2

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Square of a number sum of digits ×

having all ones in the square

2number of digits= × the number

in the number

5. Pythagorean triplet

Let us observe the sum of squares of 3 and 4

i.e., 32 + 42 = 9 + 16 = 25

32 + 42 = 52

Here, the sum of squares of 3 and 4 is square of another number 5.

Since the sum of squares of two numbers is again square of another number,

these numbers (3, 4, 5) are said to form a Pythagorean triplet.

In general, a triplet (m, n and p ) of natural numbers m, n and p is said to be a Pythagoreantriplet if m2 + n2 = p2.

Note:

i) If m > 1 and m is odd, then the Pythagorean triplet is 2 21 1, ,2 2

m mm .

Example:

If one of the numbers of a Pythagorean triplet is 3, find the triplet.

Sol.As the given number 3 is odd,2 21 1, ,2 2

m mm

i.e., m = 3

The Pythagorean triplet is 2 21 1, ,2 2

m mm2 23 1 3 1i.e, 3, ,2 2

8 10i.e, 3, ,2 2

5,4,3,e.i

Thus, the required triplet is (3, 4, 5).

Note:

ii) If m > 1 and 2m is one number of the triplet then the Pythagorean triplet is (2m, m2

– 1, m2 + 1).

Example:

If one of the numbers of a Pythagorean triplet is 4, find the triplet.

Sol: As the given number 4 is even

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i.e., 2m = 4 m = 4/2 = 2, m = 2

The Pythagorean triplet is (2m, m2 – 1, m2 + 1)

i.e., (2(2), 22 – 1 , 22 + 1)

i.e., (4,3,5)

Thus, the required triplet is (3,4,5).

Finding the Square of a Number:

Square of small numbers like 3, 4, 5, 6, 7, ...... etc. are easy to find. But can we find thesquare of 23 so quickly?

The answer is not so easy and we may need to multiply 23 by 23.

There is a way to find this without having to multiply 23 23.

We know 23 = 20 + 3

Therefore 232 = (20 + 3)2 = 20 (20 + 3) + 3 (20 + 3)

= 202 + 20 x 3 + 3 x 20 + 32

= 400 + 60 + 60 + 9 = 529.

Other patterns in squares:

Consider the following pattern:

252 = 625 = (2 x 3) hundreds + 25

352 = 1225 = (3 x 4) hundreds + 25

752 = 5625 = (7 x 8) hundreds + 25

1252 = 15625 = (12 x 3) hundreds + 25

Now you can find the square of 95?

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WORKSHEET 1

1. What will be the unit digit of the squares of the following numbers?

(i) 81 (ii) 272 (iii) 799 (iv) 3853 (v) 1234 (vi) 26387 (vii) 52698

(viii) 99880 (ix) 12796 (x) 55555

2. The following numbers are obviously not perfect squares. Give reason.

(i) 1057 (ii) 23453(iii) 7928 (iv) 222222 (v) 64000 (vi) 89722 (vii) 222000

(viii) 505050

3. The squares of which of the following would be odd numbers?

(i) 431 (ii) 2826 (iii) 7779 (iv) 82004

4. Observe the following pattern and find the missing digits.

112 = 121

1012 = 10201

10012 = 1002001

1000012 = 1…2…1

100000012 = …

5. Using the given pattern, find the missing numbers.

12 + 22 + 22 = 32

22 + 32 + 62 = 72

32 + 42 + 122 = 132

42 + 52 + _ 2 = 212

52 + _ 2 + 302 = 312

62 + 72 + _ 2 = __2

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6. 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

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

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

8. How many numbers lie between squares of the following numbers?

(i) 12 and 13 (ii) 25 and 26 (iii) 99 and 100

9. Is 63504 a perfect square? If so, find the number whose square is 63504.

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

i) 213 ii) 3824 iii) 9777 iv) 40028

11. Write a Pythagorean triplet whose one member is

(i) 6 (ii) 14 (iii) 16 (iv) 18

12. Find the square of the following numbers

(i) 32 (ii) 35 (iii) 86 (iv) 93 (v) 71 (vi) 46

13. If 8 is one of the numbers in a Pythagorean triplet, then find the triplet.

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14. Complete the following table of perfect squares and their square roots:

Perfect Square Square Root

1 1

4 2

9 3

36

8

100

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Square roots

Let us consider 811/2

As 81 is raised to the power 1/2, it is said to be the square root of 81.

Denoted as 12

Exponential Radical form form

81 81

Thus, any number a raised to the power 1/2 is said to be the square root of the givennumber, denoted as a .

Finding Square root of a Number by Prime Factorization Method

In the previous topics we have seen different methods of finding the squares of numbersgiven. Let us now have a look at the various methods for finding the square root of thegiven number.

Step-1: Write the prime factorisation of the given number.

Step-2: Pair the factors such that primes in each pair are equal.

* * *

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Step-3: Choose one prime from each pair and multiply all such primes.

Step-4: The product thus obtained is the square root of the given number.

Let us understand this method through the example given below.

Example: Find the square root of 24336.

24336 (2 2) (2 2) (3 3) (13 13)

= 2 × 2 × 3 × 13 = 156

24336 156

WORKSHEET 2

1. What could be the possible ‘one’s’ digits of the square root of each of the following

numbers?

(i) 9801 (ii) 99856 (iii) 998001 (iv) 657666025

2. Without doing any calculation, find the numbers which are surely not perfect

squares.

(i) 153 (ii) 257 (iii) 408 (iv) 441

3. Find the square roots of 100 and 169 by the method of repeated subtraction.

4. Find the square roots of the following numbers by the pme factorisation method.

(i) 729 (ii) 400 (iii) 1764 (iv) 4096 (v) 7744 (vi) 9604

(vii) 5929 (viii) 9216 (ix) 529 (x) 8100

5. For each of the following numbers, find the smallest whole number by which it

should be multiplied so as to get a perfect square number. Also find the squareroot of the square number so obtained.

(i) 252 (ii) 180 (iii) 1008 (iv) 2028 (v) 1458 (vi) 768

* * *

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6. For each of the following numbers, find the smallest whole number by which it

should be divided so as to get a perfect square number. Also find the square rootof the square number so obtained.

(i) 252 (ii) 2925 (iii) 396 (iv) 2645 (v) 2800 (vi) 1620

7. The students of Class VIII of a school donated Rs 2401 in all, for Prime Minister’s

National Relief Fund. Each student donated as many rupees as the number ofstudents in the class. Find the number of students in the class.

8. 2025 plants are to be planted in a garden in such a way that each row

contains as many plants as the number of rows. Find the number of rows and thenumber of plants in each row.

9. Find the smallest square number that is divisible by each of the numbers 4, 9,

and 10.

10. Find the smallest square number that is divisible by each of the numbers 8, 15,

and 20.

11. A square has an area of 16 units2.

a. What is the side length of a square of this area?b. Draw a square of area 16 units2 below.c. What is the square root of 16?d. Explain why your answers in parts (a) and (c) are the same.

12. A checkerboard is a square made up of 32 black and 32 red squares.

Assume that each square has a side length of 1 unit.a. What is the total area of the checkerboard?b. What is the side length of the checkerboard?c. Explain how your answers in parts (a) and (b) help you determine the

square root of 64.

13. The symbol means the positive or principal square root of a number.

a. Evaluate 121 .

b. What is the negative square root of 121?

c. A square has an area of 121 units2. What is the side length of square ofthis area?

d. Explanin why the answer to part (c) can only be the positive square rootof 121.

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Long Division Method:

Let us understand this method through the following example.

Example: Find the square root of 467856.

Step-1: Placing bars over every pair of digits i.e., 4678 56 .

Step-2: Finding the largest number whose square is less than or equal to the left most bar.

i.e., 36 < 46 6 is the required divisor. Now divide 46 by 6.

4678 5636

10

6

6

Step-3: Bring down the number under the next bar, beside the remainder of Step 2.

4678 563610 78

6

6

Step-4: Doubling the quotient i.e., 2 × 6 = 12.

46 78 5636

10 78

68

6

10 2454

128

Step-5: Guessing the largest possible digit to be taken beside the divisor and also to get thenew digit in the quotient. The required digit is8. Now divide 1078 by 128 and get theremainder 54.

Step-6: Bringing down the number under the next bar to the right of the new remainder.

4678 56361078

68

6

102454 56

128

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Step-7: Repeating the steps 4, 5 and 6 till all the bars have been considered. The finalquotient is the required square root.

46 78 5636

1078

684

6

102454 56

128

136454 56

0

467856 684

Note: This method is more efficient with larger numbers, but it can also be used to findsquare root of smaller numbers i.e., 3 digit or 4 digit numbers.

Estimating Square Roots

Most calculators have a square root button that quickly calculates square roots. There areother ways of calculating square roots but they aren’t quick. If you need an approximatevalue for a square root you can use a method like the one below.

What is the square root of 42?

What two squares does 42 come between? 36 and 49

What are the square roots of 36 and 49 6 and 7

So the square root of 42 is between 6 and 7.Let’s use some trial and error to get anapproximate answer.

Let’s try 6.5 42 ÷ 6.5 = 6.46

We’re looking to get the number we divide by to be as close to the answer we get aspossible. In this case we have 6.5 and 6.46Close, but we can get closer.

Let’s take the average of 6.5 and 6.46 and try that. 42 ÷ 6.48 = 6.48

So we have an square root that is accurate to two decimal places. You can repeatthese steps to get as accurate an answer as you want.

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Finding Square Root by Assumption Method

Steps for finding the square roots up to four digit numbers, without using eitherfactorization or division methods.

Step-1: Find the largest number whose square is less than (or) equal to the numberunder the left most bar. This is the tens digit of the square root.

Step-2: Find the units digit by squaring the relative number.

Step-3: Choose the correct digit by squaring one possible square root and comparing itwith the given number.

Example: Find the square root of 9801.

Sol: 98 01

Step-1: 92 = 81 is the largest square number < 98;

The tens digit in the square root of 9801 is 9.

9801 9 ? ;

Step-2: 12 = 1 ; 92 = 8 1 in both the cases, the units digit is 1.

Step-3: Trial 912 = 8281 9801

i.e., 1 doesn’t satisfy the units digit of square root of 9801.

9801 99

Example: Find the square roots of 144 and 6561.

Sol: 144

Step-1: 1 44

12 = 1

Tens digit in square root of 144 is 1. 144 1 ?

Step-2: 22 = 4 ; 82 = 6 4

In both the cases, the units digit is 4.

Step-3: Trial 122 = 144 144 12

Example: Find the square roots of 6561.

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Sol: 6561

Step-1: 65 61

82 = 64

Tens digit in square root of 6561 is 8 . 6561 8 ?

Step-2: 12 = 1 ; 92 = 8 1

In both the cases, the units digit is 1.

Step-3: Trial 802 = 6400 is closer to 6561 than 902 = 8100

6561 is closer to 80 than 90

The tens digit of square root may be 8

Square root of 6561 may be 81

Verification : 81×81=6561

6561 81

Square roots of decimals

Step-1: Place bars on the integral part of the number in the usual manner.

Step-2: Place bars on the decimal part on every pair of digits beginning with the firstdecimal place.

Step-3: Start finding the square root by the division process as usual.

Step-4: Place decimal point in the quotient as soon as the integral part is over.

Step-5: Stop when the remainder is zero.

Step-6: The quotient at this stage is the square root.

Example: Find the square root of 0.00059049

Step-1: Here, integral part is 0. So place a bar on 0.

Step-2: Starting from the first decimal point, place the bars on every pair of digits. i.e.,

00059049 .

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Step-3: Start finding the square root and place the decimal point in the quotient assoon as the integral part is over.

0.00 0590494

1 90

0.0243

2

44

0

1 76

14491449

483

0.0005905 0.0243

Tip (i) : If the integral part is zero in the given decimal number, then the integral part inthe square root is also zero.

Tip (ii): If the first pair after the decimal point is 00, then the first digit after the decimalin the square root is also zero.

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WORKSHEET 3

1. A man arranges 15376 apple plants in his garden in an order, so that there are as

many rows as there are apple plants in each row. Find the number of rows.

2. The product of three consecutive even numbers when divided by 8 is 720. Find

the product of their square roots.

3. What least number must be subtracted from 16160 to get a perfect square? Also

find the square root of this perfect square.

4. What least number must be added to 2945 to get a perfect square number? What

is the resulting number? Find the square root of the resulting number.

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

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

7. Find the value of 0.9 up to 3 places of decimal.

8. Find the value of 3 correct to 3 places of decimal.

9. Find the value of 37 up to four decimal places

10. Find the square root of each of the following numbers by division method.

(i) 2304 (ii) 4489 (iii) 3481 (iv) 529 (v) 3249 (vi) 1369

(vii) 5776 (viii) 7921 (ix) 576 (x) 1024 (xi) 3136 (xii) 900

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11. Find each square root:

1. - 441

2. 0

3. 324

4.2581

5.25616

6. -4

484

7. 3.61

8. - 0.64

9. - 0.01

10. 274576

11. 58081

12. 152.2756

12. Estimate the square roots of the following to the closest tenth:

1. 28.28

2. 500

3. 350

4. 31.89

5. 40.04

6. 110

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Lower PerfectSquare

Square Root

Area = 75 units2

Upper PerfectSquare

Square Root

b. Using the table completed in part (a), answer the following question:

Which perfect square is closer to 75: the lower perfect square or the upperperfect square? Tick your answer below:

Lower Perfect Square or Upper Perfect Square

c. Estimate the square root of 75:

75 = (nearest tenth)

d. Can you use the same lower and upper squares to estimate the value of 89?Explain why or why not?

13. The square roots of some numbers are not whole numbers. Suppose you

construct a square of area 75 units2.

a. Fill in the blanks below:

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14. Fill in the blanks below to estimate the square roots of non-perfect squares:

a. Area = 108 units2

Lower PerfectSquare

Square Root

Area = 108 units2

Upper PerfectSquare

Square Root

108 (nearest tenth)

b. Area = 12 units2

Lower PerfectSquare

Square Root

Area = 12 units2

Upper PerfectSquare

Square Root

12 (nearest tenth)

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c. Area = 126.8 units2

Lower PerfectSquare

Square Root

Area = 126.8 units2

Upper PerfectSquare

Square Root

126.8 (nearest tenth)

d. Area = 46.2 units2

Lower PerfectSquare

Square Root

Area = 46.2 units2

Upper PerfectSquare

Square Root

46.2 (nearest tenth)

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15. Which letter on the number line below corresponds to each square root?

56 __________

10 __________

39 __________

7 __________

32 __________

98 __________