Find the square root of 144 by repeated subtraction, prime factorization and long division method
The correct answer and explanation is:
To find the square root of 144 using three different methods, here’s how it’s done:
1. Repeated Subtraction Method:
This method works by subtracting successive odd numbers from the given number (144) until you reach zero. The number of subtractions gives you the square root.
- Start with 144.
- Subtract 1 (the first odd number).
- Subtract 3 (the second odd number).
- Continue subtracting 5, 7, 9, 11, etc., increasing the odd number by 2 each time.
- After 12 subtractions, you will reach zero, and the number of subtractions (12) is the square root of 144.
Thus, the square root of 144 is 12.
2. Prime Factorization Method:
To use prime factorization, break the number into prime factors and then group them in pairs.
- Start by dividing 144 by the smallest prime, 2.
- 144 ÷ 2 = 72.
- 72 ÷ 2 = 36.
- 36 ÷ 2 = 18.
- 18 ÷ 2 = 9.
- 9 ÷ 3 = 3.
- 3 ÷ 3 = 1.
Now, the prime factorization of 144 is:
144=24×32144 = 2^4 \times 3^2
To find the square root, take the square root of each pair of prime factors:
- Square root of 242^4 is 22=42^2 = 4.
- Square root of 323^2 is 33.
Multiply these results:
4×3=124 \times 3 = 12
Thus, the square root of 144 is 12.
3. Long Division Method:
This method involves dividing the number into groups of two digits from right to left, then using successive division and subtraction.
- Group 144 as (1) (44).
- Find the largest number whose square is less than or equal to the first group (1). The answer is 1, so subtract 12=11^2 = 1 from 1. The remainder is 0.
- Bring down the next group, 44. Now, double the divisor (which is 1) to get 2.
- Find a number that, when placed next to 2, and multiplied by the same number, gives a result close to 44. This number is 2. Multiply 22 by 2 to get 44. Subtract 44 from 44. The remainder is 0.
Since no remainder is left, the square root is 12.
Thus, the square root of 144 is 12.