Find the square root of 50625 by division method show your work
The correct answer and explanation is:
Let’s find the square root of 50625 using the division (or long division) method step by step.
Step 1: Pair the digits from right to left
Write 50625 and pair the digits starting from the right:
50 | 62 | 5
So, the pairs are 50, 62, and 5.
Step 2: Find the largest number whose square is less than or equal to the first pair (50)
- 7² = 49 (less than 50)
- 8² = 64 (greater than 50)
So, the first digit of the root is 7.
Step 3: Subtract the square of 7 from 50
50 – 49 = 1
Bring down the next pair (62) next to 1, making 162.
Step 4: Double the quotient found so far (7 × 2 = 14)
This will be the divisor’s first part. Now find a digit X such that 14X × X is less than or equal to 162.
Try X=1: 141 × 1 = 141 (less than 162)
Try X=2: 142 × 2 = 284 (greater than 162)
So, X=1.
Step 5: Write 1 in the quotient, subtract 141 from 162
162 – 141 = 21
Bring down the next pair (25) to make 2125.
Step 6: Double the current quotient (71 × 2 = 142)
Now find X such that 142X × X ≤ 2125.
Try X=1: 1421 × 1 = 1421 (less than 2125)
Try X=2: 1422 × 2 = 2844 (greater than 2125)
So, X=1.
Step 7: Write 1 in the quotient, subtract 1421 from 2125
2125 – 1421 = 704
No more pairs left to bring down.
Step 8: The quotient is 711, and the remainder is 704 (less than divisor)
Since remainder is less than the divisor, the square root is 711.
Verification:
Calculate 711²:
711 × 711 = (700 + 11) × (700 + 11)
= 700² + 2 × 700 × 11 + 11²
= 490000 + 15400 + 121
= 505,521 (this is less than 50625, so check calculations)
Oops! Seems we made an error in the previous steps. Let’s revisit carefully.
Revised step after step 5:
At step 5, after subtracting 141 from 162, remainder is 21. Bring down next pair (25) makes 2125.
At step 6, double the quotient so far (71 × 2 = 142), now find X such that 142X × X ≤ 2125.
Try X=1: 1421 × 1 = 1421
Try X=2: 1422 × 2 = 2844 (too big)
X=1.
Subtract 1421 from 2125 = 704. No more pairs left.
Now, the quotient is 711.
Check 711²:
711 × 711 = ?
Let’s multiply:
711 × 711
Stepwise:
711 × 700 = 497,700
711 × 11 = 7,821
Add: 497,700 + 7,821 = 505,521
This is less than 50625. But the original number is 50625, which is 50,625, not 506,25 (note the difference in digit placement). The original number is 50625, not 506,250.
Let’s re-check the pairing:
The number is 50625 (five digits), so pairs are:
5 | 06 | 25
Pairs: 5, 06, 25
Try again with this.
Corrected Step 1: Pair the digits
Pairs: 5 | 06 | 25
Step 2: Largest square less than or equal to 5
2² = 4 (less than 5)
3² = 9 (greater than 5)
So, first digit of root = 2.
Step 3: Subtract square of 2 from 5
5 – 4 = 1
Bring down next pair 06 → 106.
Step 4: Double the quotient so far: 2 × 2 = 4
Now find digit X such that 4X × X ≤ 106.
Try X=2: 42 × 2 = 84 (less than 106)
Try X=3: 43 × 3 = 129 (greater than 106)
So, X=2.
Step 5: Write 2 in quotient, subtract 84 from 106
106 – 84 = 22
Bring down next pair 25 → 2225.
Step 6: Double quotient so far (22 × 2 = 44)
Find X such that 44X × X ≤ 2225.
Try X=4: 444 × 4 = 1776 (less than 2225)
Try X=5: 445 × 5 = 2225 (exactly equal!)
Step 7: Write 5 in quotient
Subtract 2225 – 2225 = 0
No remainder, so the quotient is 225.
Final answer:
50625=225\sqrt{50625} = 225
Explanation (300 words):
The division method, also called the long division method, is a manual technique to find the square root of a number by repeated guessing and subtracting. The number is divided into pairs of digits starting from the decimal point towards the left and right. For the number 50625, pairs are formed as 5 | 06 | 25.
The process begins by finding the largest square number less than or equal to the first pair. For 5, it is 2² = 4, so 2 is the first digit of the root. Subtract 4 from 5, leaving a remainder of 1. The next pair (06) is then brought down to join the remainder, making 106.
Next, double the current quotient (2 × 2 = 4) and find a digit X such that when you append X to 4 (forming 4X) and multiply by X, the result is less than or equal to 106. Trying 2, 3, and so on, shows that 42 × 2 = 84 fits, so 2 is appended to the quotient, now 22. Subtract 84 from 106 to get 22.
Bring down the next pair (25) to make 2225. Double the quotient so far (22 × 2 = 44) and find X such that 44X × X ≤ 2225. Testing digits shows 445 × 5 = 2225 fits perfectly, so append 5 to the quotient, making it 225.
Subtracting 2225 from 2225 leaves zero remainder, so the process ends. The square root of 50625 is therefore 225.
This method ensures a precise calculation without guessing or estimation, making it reliable especially before calculators were common. It systematically builds the root digit by digit by combining division, multiplication, and subtraction operations.