Find the square root of 175.
To find the square root of 175, follow these steps:
Step 1: Start by estimating a number that might be the square root of 175. Since 13 x 13 = 169, it's reasonable to estimate that the square root of 175 should be slightly greater than 13.
Step 2: Take your estimate (13) and square it: 13 x 13 = 169.
Step 3: Notice that 169 is less than 175, so we need to adjust our estimate.
Step 4: Now, increase the estimate by a small amount and try again. Let's try adding 1 to our estimate: 13 + 1 = 14.
Step 5: Square the updated estimate: 14 x 14 = 196.
Step 6: Notice that 196 is greater than 175, so we need to adjust our estimate again.
Step 7: Now, decrease the estimate by a small amount. Let's try subtracting 0.1 from our estimate: 13.9.
Step 8: Square the updated estimate: 13.9 x 13.9 = 193.21.
Step 9: Notice that 193.21 is still greater than 175. We need to continue adjusting our estimate.
Step 10: Repeat steps 7-9, adjusting the estimate until you find a value close enough to the square root of 175.
After following these steps, we can determine that the square root of 175 is approximately 13.228.
To find the square root of 175, you can use one of the following methods:
Method 1: Using a calculator
1. Turn on your calculator.
2. Enter the number 175.
3. Press the square root (√) button.
4. Read the result displayed on your calculator screen.
Method 2: Using prime factorization
1. Start by finding the prime factorization of 175.
- Divide 175 by the smallest prime number, which is 2. Since it is not divisible evenly, move to the next prime number, which is 3.
- Divide 175 by 3: 175 ÷ 3 = 58 remainder 1.
- Since 58 is not divisible evenly by any prime numbers, move to the next prime number, which is 5.
- Divide 58 by 5: 58 ÷ 5 = 11 remainder 3.
- Finally, 11 is a prime number itself.
2. Write down the prime factorization of 175 as a product of powers of prime numbers:
175 = 5 × 5 × 7
3. Since we are looking for the square root, we can pair the prime factors in pairs:
√(175) = √(5 × 5 × 7) = √(5^2 × 7) = √(5^2) × √(7) = 5√(7).
Therefore, the square root of 175 is 5√(7).