How to Find Square Root of 225 Without Calculator

Method 1: Prime Factorization

Prime factorization is a systematic way to find square roots by breaking down the number into its prime components. Here's how to do it for 225:

1. Start by dividing 225 by the smallest prime number, which is 3:
   • 225 ÷ 3 = 75
   • 75 ÷ 3 = 25
   • 25 ÷ 3 = 25 (doesn't divide evenly)
2. Now divide by the next smallest prime number, which is 5:
   • 25 ÷ 5 = 5
   • 5 ÷ 5 = 1
3. Now write the prime factors:
   • 225 = 3 × 3 × 5 × 5
4. Pair the prime factors:
   • (3 × 5) × (3 × 5) = 15 × 15
5. Take one number from each pair to find the square root:
   • √225 = 15

This method works well for perfect squares, which are numbers that are squares of integers. 225 is a perfect square (15 × 15), so this method gives us the exact answer.

Method 2: Estimation

Estimation is a quick method that works well for numbers you know are close to perfect squares. Here's how to estimate the square root of 225:

1. Identify perfect squares near 225:
   • 14² = 196
   • 15² = 225
   • 16² = 256
2. Since 225 is exactly 15², the square root is clearly 15.

This method is fastest when you're dealing with numbers you know are perfect squares, but it can also be used for estimation with other numbers.

Method 3: Long Division

Long division is a more complex method that can be used for any number, including non-perfect squares. Here's how to find √225 using long division:

1. Write 225 as 225.000000 (add decimal places for precision).
2. Find the largest number whose square is less than or equal to 225:
   • 14² = 196
   • 15² = 225
3. Since 15² = 225, we've found our exact square root.
4. For demonstration, let's show the process for a non-perfect square like 224:
   • 14² = 196 (subtract from 224 to get 28)
   • Bring down two zeros (2800)
   • 140 × 2 = 280 (subtract to get 2520)
   • Bring down two zeros (252000)
   • 1400 × 7 = 9800 (subtract to get 252000 - 9800 = 242200)
   • This process would continue for more decimal places

For 225, the long division method confirms that the exact square root is 15, as we already knew from the other methods.

Verification

To ensure our answer is correct, we can verify by squaring our result:

Since 15 squared equals 225, our calculation is correct. This verification step is important for understanding and confirming our results.