Question

What is the square root of 506.25 using a long division method?
(a). 22.5
(b). 21.4
(c). 23.4
(d). 25.3

Answer 1

Hint: A square root of a number is a value that, when multiplied by itself, gives the number. Recall the method to find the square root by long division and find the value of the square root of 506.25.

Complete step-by-step solution -
The square root of a number is another number that produces the first number when it is multiplied by itself. For example, the square root of 25 is 5 because 5 multiplied with 5 is 25.
Sometimes, it is obvious to find the square root of a number, usually a smaller number but other times, the number might be big or decimal and we need to use a method called long division to arrive at the answer.
In this problem, we need to find the square root of 506.25.
As the first step in long division, we need to pair the digits from the right.
Hence, we pair 2 and 5 as 25, and we pair 0 with 6 to get 06. The digit 5 has no pair.
\[5{\text{ 06}}{\text{. 25}}\]
Next, we need to find the largest integer such that its square is less than the first number.
We know that \[{2^2}\] is 4, which is the largest square number less than 5. Then, we have:

Now, we bring the next pair down and the number becomes 106. The first digit of the divisor becomes 2 + 2 = 4, we now need to find a largest digit x such that the two-digit number 4x when multiplied with x is less than 106.
We know that 42 multiplied with 2 gives 84 which is less than 106.

We now bring the next pair down and the number becomes 2225. Since there is a decimal point before this pair, we place a decimal point in the quotient. The first terms of the divisor becomes 42 + 2 = 44. We need to find a digit such that 44x multiplied with x results in 2225.
We know that 445 when multiplied with 5 gives 2225, hence, we have:

Hence, the square root of 22.5.
Hence, the correct answer is option (a).

Note: You can cross-check your answer by squaring the final answer to check if it results in 506.25. You can use the trick that any number ending with 5 when multiplied with another number ending with 5, the units place is 5.