Square and Square Root
Square and Square Root
1) Square :-
A number multiply by itself is called square of a number.
let a number be y ,
y X y = y^2
Example :-
1) 10 X 10 = 100
2) 5 X 5 = 25
3) 11 X 11 =121
2) Square Root :-
If x^2 = y, we say that square root of y is x and we write, √y = x
Rule :-
If a given number is a perfect square, we resolve it into the product of prime factors and take the product of prime factors choosing one out of every pair of the same primes.
Example :-
1) Evaluate : √225
Sol. Resolving 225 into prime factors , we get √ 225 =√( 3 X 3 X 5 X 5 )
= 3 X 5
= 15
2) Find the square root of 1734489.
Sol. 1317
I-------------------------------------
1 I 01 73 44 89
+ 1 I -1
-------- I-------------------------------------
23 I 0 73
+3 I - 69
-------- I--------------------------------------
261 I 444
+ 1 I - 261
-------- I--------------------------------------
2627 I 18389
+7 I - 18389
------------I---------------------------------------
00000
√ 1734489 = 1317
3) Write the square root of 1000 in natural number
Solution:
√ 1000
| 31.622
3 | 1000
+ 3 | - 9
61 | 100
+ 1 | - 61
626 | 3900
+ 6 | -3756
6322 | 14400
+ 2 | - 12644
63242 | 175600
+2 | - 126484
49116
Answer = 31.622
4) What is the square root of 3481 by the long division method ?
Solution :
√3481
| 59
5 | 3481
+ 5 | - 25
109 | 981
+ 9 | - 981
000
Answer = 59
3) Square root of decimal fractions :
Rule :
We make even number of decimal places by affixing a zero , if necessary .
1) Example :
Evaluate √176.252176 .
Sol. 13.276
----------------I----------------------------------------
1 I 01 76 25 21 76
+ 1 I - 1
----------------I----------------------------------------
23 I 0 76
+ 3 I - 69
----------------I-----------------------------------------
262 I 0725
+ 2 I - 524
----------------I-----------------------------------------
2647 I 20121
+ 7 I - 18529
---------------I-------------------------------------------
26546 I 159276
+ 6 I - 159276
--------------I---------------------------------------------
000000
√176.252176 = 13.276
or Alternate method
√176252176
= --------------------
√1000000
13276
= ------------------
1000
= 13.276
2) Evaluate √ 0.4 up to 3 places of decimal.
Sol. 0.4 = 0.40 (Making even decimal places)
0.632
----------------I--------------------------------------------
6 I 0.40 00 00
+ 6 I - 36
----------------I---------------------------------------------
123 I 400
+ 3 I - 369
----------------I---------------------------------------------
1262 I 3100
+ 2 I - 2524
----------------I----------------------------------------------
√0. 40 00 00 = 0.632
or Alternate method
√040 00 00
√0.40 00 00 = ------------------
√ 100 00 00
632
= --------
1000
= 0.632
3) An important formula :
1) √ab = √a X √b
2) √a/b = √a / √b
Example :-
i) √27 X √12
Sol. = √3 X 3 X 3 X 3 X 2 X 2
= 3 X 3 X 2
= 18
ii) √ (288/128 )
Sol. = √ ( 2 X 2 X 2 X 2 X 2 X 3 X 3 )/( 2 X 2 X 2 X 2 X 2 X 2 X 2 )
= 3 / 2
iii) If √21 = 4.582 , find the value of √ 3/7
Sol. = √ 3/7 = √ (3X7)/(7X7)
= √ 21/7
= 4.582/7
= 0.6545.
Exercise
1) Which of the following can not be a digit in the unit place of a perfect square ?
a) 7 b) 1 c) 5 d) 0
2) The square root of ( 7 + 2√10 ) is :
a) √5 + √ 2 b) √3 + √4 c) √6 + 1 d) 2+ √5
Answer 1) a 2) a
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