Complete the following pattern with the help of first part given below. S.No. Value of n Addition of first n odd natural numbers n 2 (i) For n = 4 1 + 3 + 5 + 7 4 2 = 16 (ii) For n = 5 _ _ _ _ _ _ _ _ _ _ (iii) For n = 6 _ _ _ _ _ _ _ _ _ _ (iv) For n = 7 _ _ _ _ _ _ _ _ _ _ (v) For n = 8 _ _ _ _ _ _ _ _ _ _

Observing the pattern here, we have For n = 5 , Addition of first 5 odd natural numbers = 1 + 3 + 5 + 7 + 9 and n 2 = 5 2 = 25 For n = 6 , Addition of first 6 odd natural numbers = 1 + 3 + 5 + 7 + 9 + 11 and n 2 = 6 2 = 36 For n = 7 , Addition of first 7 odd natural numbers = 1 + 3 + 5 + 7 + 9 + 11 + 13 and n 2 = 7 2 = 49 For n = 8 , Addition of first 8 odd natural numbers = 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 and n 2 = 8 2 = 64 Interestingly, we may note that the sum of the first n odd natural numbers is always n 2 , i.e., 1 + 3 + 5 + 7 + . . . + n th odd number = n 2 Therefore, the given table is, S.No. Value of n Addition of first n odd natural numbers n 2 (i) For n = 4 1 + 3 + 5 + 7 4 2 = 16 (ii) For n = 5 1 + 3 + 5 + 7 + 9 5 2 = 25 (iii) For n = 6 1 + 3 + 5 + 7 + 9 + 11 6 2 = 36 (iv) For n = 7 1 + 3 + 5 + 7 + 9 + 11 + 13 7 2 = 49 (v) For n = 8 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 8 2 = 64

Observe the following pattern and complete the remaining part 1 2 = 1 11 2 = 1 2 1 111 2 = 1 2 3 2 1 1111 2 = _ _ _ 4 _ _ _ 11111 2 = _ _ _ _ 5 _ _ _ _ 111111 2 = _ _ _ _ _ 6 _ _ _ _ _ 1111111 2 = _ _ _ _ _ _ 7 _ _ _ _ _ _

The numbers whose squares are given in the pattern, are 1 , 11 , 111 and so on. All the numbers have only 1 as the repeated digit, with the first number have one 1 , the second number having two 1 's, the third number having three 1 's and so on till the last number which has seven 1 's. Also, the pattern of the squares of these numbers is interesting to observe. In each of these squares, apart from the first one, the digits start from 1 , increase in steps of 1 till the central digit is reached, which happens to be equal to the number of 1 's in the given number. After the central digit, the digits decrease in steps of 1 , till the last digit 1 is reached. Thus, we have, the pattern as shown 1 2 = 1 11 2 = 1 2 1 111 2 = 1 2 3 2 1 1111 2 = 1 2 3 4 3 2 1 11111 2 = 1 2 3 4 5 4 3 2 1 111111 2 = 1 2 3 4 5 6 5 4 3 2 1 1111111 2 = 1 2 3 4 5 6 7 6 5 4 3 2 1

See the pattern given in the table and find the square of the given numbers without doing actual multiplication. S.No. Number Pattern (i) 33 2 30 + 3 2 ⇒ 30 30 + 3 + 3 30 + 3 ⇒ 30 2 + 30 × 3 + 3 × 30 + 3 2 ⇒ 900 + 90 + 90 + 9 = 1089 (ii) 43 2 40 + 3 2 = _____ = _____ = _____ (iii) 56 2 50 + 6 2 = _____ = _____ = _____

The idea here is to learn a simpler and faster method to compute the squares of numbers. This is done using the distributive property of multiplication over addition. The distributive property states that for any numbers, say a , b and c , a b + c = a b + a c As observed here, the number whose square is to be computed, is conveniently split into the addition of two numbers, one of them being a multiple of 10 and the other a single digit number. Here, 43 = 40 + 3 Thus, 43 2 = 40 + 3 2 ⇒ 43 2 = 40 40 + 3 + 3 40 + 3 ⇒ 43 2 = 40 2 + 40 × 3 + 3 × 40 + 3 2 ⇒ 43 2 = 1600 + 120 + 120 + 9 ⇒ 43 2 = 1849 Similarly, 56 = 50 + 6 Therefore, 56 2 = 50 + 6 2 ⇒ 56 2 = 50 50 + 6 + 3 50 + 6 ⇒ 56 2 = 50 2 + 50 × 6 + 6 × 50 + 6 2 ⇒ 56 2 = 2500 + 300 + 300 + 36 ⇒ 56 2 = 3136 Hence, the table is, S.No. Number Pattern (i) 33 2 30 + 3 2 ⇒ 30 30 + 3 + 3 30 + 3 ⇒ 30 2 + 30 × 3 + 3 × 30 + 3 2 ⇒ 900 + 90 + 90 + 9 = 1089 (ii) 43 2 40 + 3 2 ⇒ 40 40 + 3 + 3 40 + 3 ⇒ 40 2 + 40 × 3 + 3 × 40 + 3 2 ⇒ 1600 + 120 + 120 + 9 = 1849 (iii) 56 2 50 + 6 2 ⇒ 50 50 + 6 + 3 50 + 6 ⇒ 50 2 + 50 × 6 + 6 × 50 + 6 2 ⇒ 2500 + 300 + 300 + 36 = 3136

Complete the following table by given pattern. S.No. Number Pattern Number of digits in the square root (i) 64 For even, n 2 2 2 = 1 (ii) 125 For odd, n + 1 2 3 + 1 2 = 4 2 = 2 (iii) 144 _ _ _ _ _ _ _ _ _ _ (iv) 169 _ _ _ _ _ _ _ _ _ _ (v) 7569 _ _ _ _ _ _ _ _ _ _ (vi) 10201 _ _ _ _ _ _ _ _ _ _

Here, the pattern that is to be observed is based on the number of digits in the square number. If n is the number of digits in the square number, then we may find the number of digits in the square root as n 2 , if n is even and n + 1 2 , if n is odd. 144 has 3 digits (odd). Hence, the number of digits in its square root will be = n + 1 2 = 3 + 1 2 = 4 2 = 2 . 169 has 3 digits (odd). Hence, the number of digits in its square root will be = n + 1 2 = 3 + 1 2 = 4 2 = 2 . 7569 has 4 digits (even). Hence, the number of digits in its square root will be = n 2 = 4 2 = 2 . 10201 has 5 digits (odd). Hence, the number of digits in its square root will be = n + 1 2 = 5 + 1 2 = 6 2 = 3 . Thus, we have S.No. Number Pattern Number of digits in the square root (i) 64 For even, n 2 2 2 = 1 (ii) 125 For odd, n + 1 2 3 + 1 2 = 4 2 = 2 (iii) 144 For odd, n + 1 2 3 + 1 2 = 4 2 = 2 (iv) 169 For odd, n + 1 2 3 + 1 2 = 4 2 = 2 (v) 7569 For even, n 2 4 2 = 2 (vi) 10201 For odd, n + 1 2 5 + 1 2 = 6 2 = 3

MEDIUM

8th CBSE

IMPORTANT

Earn 100

Find the least number that must be added to $1700$ , to get a perfect square. Also, find the square root of the result.

Important Questions on Squares and Square Roots

EASY

8th CBSE

IMPORTANT

NCERT Practice Workbook Mathematics>Chapter 6 - Squares and Square Roots>Worksheet 2>Q 9

In a right angled $∆ P Q R, ∠ Q = 90 °, P Q = 6 cm$ and $Q R = 8 cm$ . If $P R = k cm$ , then find the value of $k$ .

MEDIUM

8th CBSE

IMPORTANT

NCERT Practice Workbook Mathematics>Chapter 6 - Squares and Square Roots>Worksheet 2>Q 10

Find the smallest square number that is divisible by each of the numbers

8, 15

and

20

. Also, find the square root of the result.

EASY

8th CBSE

IMPORTANT

NCERT Practice Workbook Mathematics>Chapter 6 - Squares and Square Roots>Worksheet 3>Q 1

Complete the following pattern with the help of first part given below.

S.No.	Value of $n$	Addition of first $n$ odd natural numbers	$n^{2}$
(i)	For $n = 4$	$1 + 3 + 5 + 7$	$4^{2} = 16$
(ii)	For $n = 5$	$_____$	$_____$
(iii)	For $n = 6$	$_____$	$_____$
(iv)	For $n = 7$	$_____$	$_____$
(v)	For $n = 8$	$_____$	$_____$

EASY

8th CBSE

IMPORTANT

NCERT Practice Workbook Mathematics>Chapter 6 - Squares and Square Roots>Worksheet 3>Q 3

Observe the following pattern and complete the remaining part

$1^{2}$	$=$	$1$
$11^{2}$	$=$	$1 2 1$
$111^{2}$	$=$	$1 2 3 2 1$
$1111^{2}$	$=$	$___4___$
$11111^{2}$	$=$	$____5____$
$111111^{2}$	$=$	$_____6_____$
$1111111^{2}$	$=$	$______7______$

EASY

8th CBSE

IMPORTANT

NCERT Practice Workbook Mathematics>Chapter 6 - Squares and Square Roots>Worksheet 3>Q 3

Check the pattern given below and fill in the blank.
Pattern: On adding $1$ to the product of two consecutive odd natural numbers, it gives the square of the even number.

$1 \times 3 + 1 =$ _____ $= 2^{2}$

EASY

8th CBSE

IMPORTANT

NCERT Practice Workbook Mathematics>Chapter 6 - Squares and Square Roots>Worksheet 3>Q 3

Check the pattern given below and fill in the blank.

Pattern: On adding $1$ to the product of two consecutive odd natural numbers, it gives the square of the even number.

$(2 n - 1) \times (2 n + 1) + 1 =$ $4 n^{k}$ , the value of $k =$ _____

EASY

8th CBSE

IMPORTANT

NCERT Practice Workbook Mathematics>Chapter 6 - Squares and Square Roots>Worksheet 3>Q 4

See the pattern given in the table and find the square of the given numbers without doing actual multiplication.

S.No.	Number	Pattern
(i)	$33^{2}$	${(30 + 3)}^{2}$	$\Rightarrow 30 (30 + 3) + 3 (30 + 3)$ $\Rightarrow 30^{2} + 30 \times 3 + 3 \times 30 + 3^{2}$ $\Rightarrow 900 + 90 + 90 + 9 = 1089$
(ii)	$43^{2}$	${(40 + 3)}^{2}$	$=$ _____ $=$ _____ $=$ _____
(iii)	$56^{2}$	${(50 + 6)}^{2}$	$=$ _____ $=$ _____ $=$ _____

EASY

8th CBSE

IMPORTANT

NCERT Practice Workbook Mathematics>Chapter 6 - Squares and Square Roots>Worksheet 3>Q 5

Complete the following table by given pattern.

S.No.	Number	Pattern	Number of digits in the square root
(i)	$64$	For even, $\frac{n}{2}$	$\frac{2}{2} = 1$
(ii)	$125$	For odd, $\frac{(n + 1)}{2}$	$\frac{3 + 1}{2} = \frac{4}{2} = 2$
(iii)	$144$	$_____$	$_____$
(iv)	$169$	$_____$	$_____$
(v)	$7569$	$_____$	$_____$
(vi)	$10201$	$_____$	$_____$

Find the least number that must be added to 1700, to get a perfect square. Also, find the square root of the result.

Important Questions on Squares and Square Roots

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Find the least number that must be added to $1700$ , to get a perfect square. Also, find the square root of the result.