Final answer:
To find the square roots using the long division method, we group the digits in pairs (starting from the right) and find the closest approximation in each step. The square root of the given numbers can be calculated as follows:
Step-by-step explanation:
a) √53361:
To find the square root of 53361 by long division method, we start by grouping the digits in pairs from right to left. Since 361 is the largest perfect square that is less than 533, we write 19 (the square root of 361) as the first digit. Then, we subtract 361 from 533 to get 172. We bring down the next pair of digits (2) and double the first digit of the quotient. We get 38 when we multiply 19 by 2. We find the largest digit (x) to fit as the next digit on the quotient so that 2(190 + x)x is the closest approximation to 17202. After subtracting 2(190 + x)x from 17202 and bringing down the next pair of digits (02), we repeat the process until we reach the desired level of accuracy. In this case, the square root of 53361 is approximately 231.5.
b) √0.002809:
To find the square root of 0.002809 by long division method, we start by multiplying the decimal by 10000 to make it a whole number. Then, we proceed as we did in part a. The square root of 2809 is 53. Therefore, the square root of 0.002809 is approximately 0.053.
c) √256:
The square root of 256 is 16, since 16 x 16 equals 256.
d) √1:
The square root of 1 is 1, since 1 x 1 equals 1.