Evaluate square root of 50625 by long division method ?
The correct answer and explanation is:
To find the square root of 50625 by the long division method, follow these steps carefully:
Step 1: Group the digits in pairs from right to left
50625 is grouped as 5 | 06 | 25.
Step 2: Find the largest number whose square is less than or equal to the first group (5)
The largest number is 2 because 22=4≤52^2 = 4 \leq 5.
Write 2 above the division bar.
Subtract 44 from 5, remainder is 1.
Step 3: Bring down the next pair (06), making the new dividend 106.
Step 4: Double the number already found (2) and write it as 4_ (40 plus a digit to find)
Now find a digit xx such that 4x×x≤1064x \times x \leq 106.
Try x=2x=2: 42×2=84≤10642 \times 2 = 84 \leq 106
Try x=3x=3: 43×3=129>10643 \times 3 = 129 > 106
So, x=2x=2. Write 2 next to 2 on top, making 22.
Subtract 84 from 106, remainder is 22.
Step 5: Bring down the next pair (25), making the new dividend 2225.
Step 6: Double the number on top (22), giving 44_ (440 plus a digit to find)
Find xx such that 44x×x≤222544x \times x \leq 2225.
Try x=5x=5: 445×5=2225≤2225445 \times 5 = 2225 \leq 2225 perfect!
Write 5 next to 22 on top, making 225.
Subtract 2225 from 2225, remainder is 0.
Since the remainder is zero and all digit pairs have been processed, the square root is exactly 225.
Explanation of the Long Division Method for Square Roots
The long division method of finding square roots breaks the number into pairs of digits starting from the decimal point or right end if the number is an integer. Each pair is treated as a chunk to work through step-by-step.
The method works by guessing digits of the root one by one, starting from the left. At each step, the current root approximation is doubled and used to form a divisor to estimate the next digit. The process is similar to regular division, but with multiplication involving the root estimate and the next digit guess.
By grouping digits into pairs, the method effectively handles the number’s scale. Subtracting the square of the root portion at each step helps to find the next digit to approximate the root more precisely. The zero remainder at the end confirms that the number is a perfect square, and the root found is exact.
This method is especially useful before calculators were common and helps to understand how square roots are built digit by digit. It can be used for decimals too by pairing digits after the decimal point.
Final answer:
225\boxed{225}