Page 59 - Start Up Mathematics_8 (Non CCE)

2
2
2
7. Observe the following pattern and find the value of: (a) 45 – 44 (b) 132 – 131 2
2
2
2 – 1 = 2 + 1
2
2
3 – 2 = 3 + 2
2
2
4 – 3 = 4 + 3
2
2
5 – 4 = 5 + 4
8. Write a Pythagorean triplet whose members are formed by the given natural numbers.
(a) 6 (b) 14 (c) 16 (d) 8
9. Which of the following numbers are squares of even numbers?
(a) 1,225 (b) 256 (c) 2,704 (d) 841 (e) 2,304
10. Using a suitable pattern, evaluate the following:
2
2
2
2
(a) (999999) (b) (99999999) (c) (111111) (d) (66666667) (e) (6667) 2
11. Give reasons why the following numbers are not perfect squares.
(a) 23,647 (b) 14,080 (c) 7,98,000 (d) 45,00,000 (e) 10,583
Shortcut Methods for Squaring a Number
1. Column Method
x
y
Let there be a two-digit number in the form   , say, 38 where x = 3 and y = 8.
1st part 2nd part
2
2
2
The rule used is (x + y) = x + 2xy + y . Remember
2
2
Thus, (3 + 8) = 3 + 2 × 3 × 8 + 8 2 Column method is easy and
2
Step 1: Make 3 columns namely I, II, III with headings x , 2xy time saving, only for two-digit
2
2
2
and y . Write the values of x , 2xy and y in columns I, II numbers.
and III, respectively.
2
Step 2: In column III, encircle the units digit of y and carry over its tens digit to column II. Add this to
the value of 2xy.
2
If y is a single digit number, nothing is to be carried to column II.
Step 3: In column II, encircle the units digit of the sum earlier obtained and I II III
2
carry over the tens digit to column I. Add this to the value of x and x 2 2xy y 2
encircle the result.
The answer is obtained by writing the encircled numbers starting 9 48 64
from column I.
I II III
encircled number encircled number encircled number x 2 2xy y 2
in column I in column II in column III
   9 48 6 4
1st part 2nd part 3rd part
+6
14 4 4 54
1st part 2nd part 3rd part
2
\ (38) = 1,444 I II III
x 2 2xy y 2
2. Observation Method 9 48 6 4
2
2
2
The rule used is (x + y) = x + 2xy + y . The idea is to observe the number and +5 +6
try to break it into smaller numbers whose squares are already memorized or are 14 5 4
easy to find.
So, if we wish to find the square of 108, first we write 108 as 108 = 100 + 8.

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