Square Root EXERCISE 1.1.1

1. Find the square root of the following numbers by the factorization method

(i) 82944

Solution:

82944 = 2¹⁰  x 3⁴

= (2⁵)²   x (3²)²

√(82944) = √((2⁵)²   x (3²)² ) = 2⁵ x 3² = 288

(ii) 155236

Solution:

155236 = (2)² x  (197)²

√(155236) = √((2)²   x (197)² ) = 2 x 197 = 394

(iii) 19881

Solution:

19881 = (3)² x  (47)²

√(19881) = √((3)²   x (47)² ) = 2 x 47 = 141

2. Find the square root of the following numbers.
(i) 184.96

Solution:

^{= (184.96X100)}⁄_{100 = ^(18496)⁄100} 

= ^{((2³)²x(17)²)}⁄_10²

_{√(184.96) = √ ^{((2³)²x(17)²}⁄10²) = ^(2³)x(17)⁄10}  
= ⁽¹³⁶⁾⁄₁₀

= 13.6

(iii) 19.5364

Solution:

19.5364 = ^{(19.5364×10000)}⁄₁₀₀₀₀
= ^(195364)⁄₁₀₀₀₀

_{√(195364/10000) = √ ^{((2)²x(221)²}⁄100²) = ^(2)x(221)⁄100}  
= ⁽⁴⁴²⁾⁄₁₀₀

= 4.42

3. Exploration: Find the squares of the numbers 9, 10, 99, 100, 1000, and 9999.Tabulate these numbers. How many digits are there in the squares of a numbers when it has even number of digits and odd number of digits?

Solution:

9² = 81; digits-2
10² = 100; digits-3
99² = 9801; digits-4
100² = 10000; digits-5
990² = 998001; digits-6
1000² = 1000000; digits-7
9999² = 99980001; digits-8

They follow the rule
2n if the number has even digits
2n-1 if the number has odd digits.
4. If a perfect square A has A digits, how many digits to you expect in √𝐀

Solution:
If a perfect square A has n digits then, √A has

⁽ⁿ⁾⁄₂ Digits if n is even
⁽ⁿ⁺¹⁾⁄₂ Digits if n is odd.