Page 30 - SM inner class 8.cdr

Estimating square roots of non perfect Thus, 5 is the smallest required number
square numbers and 2645 ¸ 5 = 529
Also 529 = 23
So far we have learnt the method for finding
the square roots of perfect squares. If the Example 20. Find the least number by which
numbers are not perfect squares, then we will 52 should be multiplied to make it a perfect
not be able to find the exact square roots. In all square.
such cases we atleast need to estimate the Solution: By prime factorization we get
square root.
2 52
For ex am ple:
2 26
Let us estimate the value of 300 to the nearest
whole number. 13 13
1
300 lies between two perfect square numbers
´
100 and 400 ....- 52 = 2 ´ 2 13
100 < 300 < 400 It is clear that in the prime factorization of
2
2
10 < 300 < 20 i.e. 52, the prime factor 13 is left unpaired.
10 < 300 < 20 \ Multiplication by 13 will make the number
a perfect square.
But still we are not very close to the square
´
´
number. Thus, 52 13´ = 2 ´ 2 13 13 is a perfect square.
2
2
,
we know that 17 = 289 18 = 324 Therefore Example 21. In a cinema hall, the number of
289 < 300 < 324 rows is equal to the number of seats in each
17 < 300 < 18 row. If the capacity of the cinema hall is 2401.
Find the number of seats in each row.
As 289 is, more closer to 300 than 324. The
approximate value of 300 is 17. Solution: Let the number of seats in each row
be x
Example 19. Find the smallest number by
which 2645 should be divided so that the According to the questions
2
quotient is a perfect square. Also find the x = 2401
square root of the quotient. x = 2401
Solution: By prime factorization we get By prime factorization we get

5 2645 7 2401
23 529 7 343
23 23 7 49
1 7 7
1
2645 = 5 ´ 23 ´ 23
It is clear that in the prime factorization of 2401 = 7 ´ 7 ´ 7 ´ 7
2645, the prime factor 5 is left unpaired.
2401 = (7 ´ ) 49
7
=
So, if we divide 2645 by 5 we get a perfect
square. Hence, the number of seats in each row = 49.
E xercise 3.2

1. Find the square root of the following by prime factorization:
(i) 289 (ii) 1225 (iii) 3844 (iv) 4225
(v) 5184 (vi) 8281 (vii) 10404 (viii) 29241
(ix) 103041 (x) 418609 (xi) 10329796 (xii) 30349081

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Mathematics In Focus - 8