Question
b) Let’s take a number 260 .
(i) Find the square root of 260 by division method.
(ii) What is the remainder left in this process?
(iii) Subtract the remainder from 260 . Is the difference a perfect square?
(i) Find the square root of 260 by division method.
(ii) What is the remainder left in this process?
(iii) Subtract the remainder from 260 . Is the difference a perfect square?
Ask by Warner Burns. in Nepal
Dec 26,2024
Upstudy AI Solution
Tutor-Verified Answer
Answer
- (i) The square root of 260 is approximately 16 with a remainder of 4.
- (ii) The remainder is 4.
- (iii) Subtracting the remainder from 260 gives 256, which is a perfect square.
Solution
Let’s solve each part of the problem step by step.
b) Given Number: 260
(i) Find the square root of 260 by the division method.
The division method for finding square roots involves a process similar to long division. Here’s how to apply it to find √260:
-
Pair the Digits:
- Start from the decimal point and pair the digits in twos: 2 | 60
-
Find the Largest Square ≤ First Pair (2):
- The largest integer whose square is ≤ 2 is 1 (since
). - Quotient so far: 1
- Remainder:
- The largest integer whose square is ≤ 2 is 1 (since
-
Bring Down the Next Pair (60):
- Combine the remainder with the next pair:
(remainder) becomes 160.
- Combine the remainder with the next pair:
-
Double the Current Quotient and Find the Next Digit:
- Double the current quotient (1):
. - Find a digit x such that
. - Testing x = 6:
- Next digit: 6
- Testing x = 6:
- Quotient so far: 16
- Remainder:
- Double the current quotient (1):
-
(Optional) Continue for Decimal Precision:
- If you need more decimal places, continue the process by bringing down pairs of zeros and repeating the steps. For simplicity, we’ll stop here.
Approximate Square Root:(to the nearest integer)
(ii) What is the remainder left in this process?
- Remainder: 4
(iii) Subtract the remainder from 260. Is the difference a perfect square?
-
Calculate the Difference:
-
Check if 256 is a Perfect Square:Yes, 256 is a perfect square since
.
Summary:
- (i) The square root of 260 is 16 with a remainder.
- (ii) The remainder is 4.
- (iii) Subtracting the remainder from 260 gives 256, which is a perfect square (
).
Answered by UpStudy AI and reviewed by a Professional Tutor
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Beyond the Answer
To find the square root of 260 using the division method, pair the digits starting from the decimal point. The pairs are (2)(60). The largest square less than or equal to 2 is 1 (1x1), so write 1 (the divisor) under 2, subtract it to get 1, and bring down the next pair (60) making it 160. Now double the 1 (making it 2) and find a digit ‘x’ such that 2x * x <= 160. You find x = 7 (27 * 7 = 189 exceeds it). So we try 6. 26 * 6 = 156 gives a valid result. Subtract 156 from 160 to get a remainder of 4.
After subtracting the remainder (4) from 260, we get 256, which is indeed a perfect square since 16 x 16 = 256! So, yay for math celebrations! 🎉