Find the square root of the following by finding the digit and the quotient. 824
The Correct Answer and Explanation is:
To find the square root of 824, we’ll use the method of long division. Let’s break it down step by step:
Step 1: Group the digits
Start by grouping the number into pairs of digits from right to left. In this case, 824 is grouped as:8 248 \, 24824
Step 2: Find the largest number whose square is less than or equal to 8
Now, we look at the first group (8). The largest number whose square is less than or equal to 8 is 2, because 22=42^2 = 422=4, and 32=93^2 = 932=9 (which is too large). So, we write 2 as the first digit of our square root.
Step 3: Subtract and bring down the next group
Now subtract 22=42^2 = 422=4 from 8:8−4=48 – 4 = 48−4=4
Now, bring down the next pair of digits (24) next to 4, making it 424.
Step 4: Double the current digit and find the next digit
Now, double the 2 (the first digit of our root) to get 4. Write it down as the divisor and leave a space next to it:4_(We will find the next digit here)4 \_ \quad \text{(We will find the next digit here)}4_(We will find the next digit here)
Now, guess a digit (let’s call it ddd) that will complete the divisor and be multiplied by ddd such that 4d×d4d \times d4d×d is less than or equal to 424. Try 8:48×8=38448 \times 8 = 38448×8=384
So, the next digit is 8. Now, we subtract:424−384=40424 – 384 = 40424−384=40
Step 5: The final square root approximation
At this stage, we have found that the square root of 824 is approximately 28.8 (since the remainder is 40, and we would continue the process to get more precise digits).
So, the square root of 824 is approximately:824≈28.8\sqrt{824} \approx 28.8824≈28.8
This method is often used when you need a quick approximation, and while this result isn’t exact, it is close enough for many practical purposes. For more precision, you would continue with more steps of long division.
