Find the square root of the number by using the method of prime factorisation:
225
441
729
1296
2025
4096
7056
8100
9216
11025
15876
17424

Answer

To find the square roots of the given numbers using the method of prime factorization, we will break down each number into its prime factors, group the factors into pairs, and then calculate the square root. Let's solve each number step by step.

### 1. Finding the square root of 225

**Step 1:** Prime factorization of 225.
- 225 is divisible by 3:
- \( 225 \div 3 = 75 \)
- 75 is also divisible by 3:
- \( 75 \div 3 = 25 \)
- 25 is divisible by 5:
- \( 25 \div 5 = 5 \)
- Finally, 5 is a prime number.

So, the prime factorization of 225 is:
\[ 225 = 3^2 \times 5^2 \]

**Step 2:** Pair the prime factors.
- We can pair the factors as follows:
- \( (3 \times 3) \) and \( (5 \times 5) \)

**Step 3:** Take one factor from each pair.
- The square root of 225 is:
\[ \sqrt{225} = 3 \times 5 = 15 \]

### 2. Finding the square root of 441

**Step 1:** Prime factorization of 441.
- 441 is divisible by 3:
- \( 441 \div 3 = 147 \)
- 147 is divisible by 3:
- \( 147 \div 3 = 49 \)
- 49 is divisible by 7:
- \( 49 \div 7 = 7 \)
- Finally, 7 is a prime number.

So, the prime factorization of 441 is:
\[ 441 = 3^2 \times 7^2 \]

**Step 2:** Pair the prime factors.
- We can pair the factors as follows:
- \( (3 \times 3) \) and \( (7 \times 7) \)

**Step 3:** Take one factor from each pair.
- The square root of 441 is:
\[ \sqrt{441} = 3 \times 7 = 21 \]

### 3. Finding the square root of 729

**Step 1:** Prime factorization of 729.
- 729 is divisible by 3:
- \( 729 \div 3 = 243 \)
- 243 is divisible by 3:
- \( 243 \div 3 = 81 \)
- 81 is divisible by 3:
- \( 81 \div 3 = 27 \)
- 27 is divisible by 3:
- \( 27 \div 3 = 9 \)
- 9 is divisible by 3:
- \( 9 \div 3 = 3 \)
- Finally, 3 is a prime number.

So, the prime factorization of 729 is:
\[ 729 = 3^6 \]

**Step 2:** Pair the prime factors.
- We can pair the factors as follows:
- \( (3 \times 3) \), \( (3 \times 3) \), \( (3 \times 3) \)

**Step 3:** Take one factor from each pair.
- The square root of 729 is:
\[ \sqrt{729} = 3^3 = 27 \]

### 4. Finding the square root of 1296

**Step 1:** Prime factorization of 1296.
- 1296 is divisible by 2:
- \( 1296 \div 2 = 648 \)
- 648 is divisible by 2:
- \( 648 \div 2 = 324 \)
- 324 is divisible by 2:
- \( 324 \div 2 = 162 \)
- 162 is divisible by 2:
- \( 162 \div 2 = 81 \)
- 81 is divisible by 3:
- \( 81 \div 3 = 27 \)
- 27 is divisible by 3:
- \( 27 \div 3 = 9 \)
- 9 is divisible by 3:
- \( 9 \div 3 = 3 \)
- Finally, 3 is a prime number.

So, the prime factorization of 1296 is:
\[ 1296 = 2^4 \times 3^4 \]

**Step 2:** Pair the prime factors.
- We can pair the factors as follows:
- \( (2 \times 2) \), \( (2 \times 2) \), \( (3 \times 3) \), \( (3 \times 3) \)

**Step 3:** Take one factor from each pair.
- The square root of 1296 is:
\[ \sqrt{1296} = 2^2 \times 3^2 = 4 \times 9 = 36 \]

### 5. Finding the square root of 2025

**Step 1:** Prime factorization of 2025.
- 2025 is divisible by 5:
- \( 2025 \div 5 = 405 \)
- 405 is divisible by 3:
- \( 405 \div 3 = 135 \)
- 135 is divisible by 3:
- \( 135 \div 3 = 45 \)
- 45 is divisible by 3:
- \( 45 \div 3 = 15 \)
- 15 is divisible by 3:
- \( 15 \div 3 = 5 \)
- Finally, 5 is a prime number.

So, the prime factorization of 2025 is:
\[ 2025 = 5^2 \times 3^4 \]

**Step 2:** Pair the prime factors.
- We can pair the factors as follows:
- \( (5 \times 5) \), \( (3 \times 3) \), \( (3 \times 3) \)

**Step 3:** Take one factor from each pair.
- The square root of 2025 is:
\[ \sqrt{2025} = 5 \times 3^2 = 5 \times 9 = 45 \]

### 6. Finding the square root of 4096

**Step 1:** Prime factorization of 4096.
- 4096 is divisible by 2:
- \( 4096 \div 2 = 2048 \)
- 2048 is divisible by 2:
- \( 2048 \div 2 = 1024 \)
- 1024 is divisible by 2:
- \( 1024 \div 2 = 512 \)
- 512 is divisible by 2:
- \( 512 \div 2 = 256 \)
- 256 is divisible by 2:
- \( 256 \div 2 = 128 \)
- 128 is divisible by 2:
- \( 128 \div 2 = 64 \)
- 64 is divisible by 2:
- \( 64 \div 2 = 32 \)
- 32 is divisible by 2:
- \( 32 \div 2 = 16 \)
- 16 is divisible by 2:
- \( 16 \div 2 = 8 \)
- 8 is divisible by 2:
- \( 8 \div 2 = 4 \)
- 4 is divisible by 2:
- \( 4 \div 2 = 2 \)
- Finally, 2 is a prime number.

So, the prime factorization of 4096 is:
\[ 4096 = 2^{12} \]

**Step 2:** Pair the prime factors.
- We can pair the factors as follows:
- \( (2 \times 2) \) for 6 pairs.

**Step 3:** Take one factor from each pair.
- The square root of 4096 is:
\[ \sqrt{4096} = 2^6 = 64 \]

### 7. Finding the square root of 7056

**Step 1:** Prime factorization of 7056.
- 7056 is divisible by 2:
- \( 7056 \div 2 = 3528 \)
- 3528 is divisible by 2:
- \( 3528 \div 2 = 1764 \)
- 1764 is divisible by 2:
- \( 1764 \div 2 = 882 \)
- 882 is divisible by 2:
- \( 882 \div 2 = 441 \)
- 441 is divisible by 3:
- \( 441 \div 3 = 147 \)
- 147 is divisible by 3:
- \( 147 \div 3 = 49 \)
- 49 is divisible by 7:
- \( 49 \div 7 = 7 \)
- Finally, 7 is a prime number.

So, the prime factorization of 7056 is:
\[ 7056 = 2^4 \times 3^2 \times 7^2 \]

**Step 2:** Pair the prime factors.
- We can pair the factors as follows:
- \( (2 \times 2) \), \( (2 \times 2) \), \( (3 \times 3) \), \( (7 \times 7) \)

**Step 3:** Take one factor from each pair.
- The square root of 7056 is:
\[ \sqrt{7056} = 2^2 \times 3 \times 7 = 4 \times 3 \times 7 = 84 \]

### 8. Finding the square root of 8100

**Step 1:** Prime factorization of 8100.
- 8100 is divisible by 2:
- \( 8100 \div 2 = 4050 \)
- 4050 is divisible by 2:
- \( 4050 \div 2 = 2025 \)
- 2025 is divisible by 5:
- \( 2025 \div 5 = 405 \)
- 405 is divisible by 3:
- \( 405 \div 3 = 135 \)
- 135 is divisible by 3:
- \( 135 \div 3 = 45 \)
- 45 is divisible by 3:
- \( 45 \div 3 = 15 \)
- 15 is divisible by 3:
- \( 15 \div 3 = 5 \)
- Finally, 5 is a prime number.

So, the prime factorization of 8100 is:
\[ 8100 = 2^2 \times 3^4 \times 5^2 \]

**Step 2:** Pair the prime factors.
- We can pair the factors as follows:
- \( (2 \times 2) \), \( (3 \times 3) \), \( (3 \times 3) \), \( (5 \times 5) \)

**Step 3:** Take one factor from each pair.
- The square root of 8100 is:
\[ \sqrt{8100} = 2 \times 3^2 \times 5 = 2 \times 9 \times 5 = 90 \]

### 9. Finding the square root of 9216

**Step 1:** Prime factorization of 9216.
- 9216 is divisible by 2:
- \( 9216 \div 2 = 4608 \)
- 4608 is divisible by 2:
- \( 4608 \div 2 = 2304 \)
- 2304 is divisible by 2:
- \( 2304 \div 2 = 1152 \)
- 1152 is divisible by 2:
- \( 1152 \div 2 = 576 \)
- 576 is divisible by 2:
- \( 576 \div 2 = 288 \)
- 288 is divisible by 2:
- \( 288 \div 2 = 144 \)
- 144 is divisible by 2:
- \( 144 \div 2 = 72 \)
- 72 is divisible by 2:
- \( 72 \div 2 = 36 \)
- 36 is divisible by 2:
- \( 36 \div 2 = 18 \)
- 18 is divisible by 2:
- \( 18 \div 2 = 9 \)
- 9 is divisible by 3:
- \( 9 \div 3 = 3 \)
- Finally, 3 is a prime number.

So, the prime factorization of 9216 is:
\[ 9216 = 2^{10} \times 3^2 \]

**Step 2:** Pair the prime factors.
- We can pair the factors as follows:
- \( (2 \times 2) \) for 5 pairs and \( (3 \times 3) \)

**Step 3:** Take one factor from each pair.
- The square root of 9216 is:
\[ \sqrt{9216} = 2^5 \times 3 = 32 \times 3 = 96 \]

### 10. Finding the square root of 11025

**Step 1:** Prime factorization of 11025.
- 11025 is divisible by 3:
- \( 11025 \div 3 = 3675 \)
- 3675 is divisible by 3:
- \( 3675 \div 3 = 1225 \)
- 1225 is divisible by 5:
- \( 1225 \div 5 = 245 \)
- 245 is divisible by 5:
- \( 245 \div 5 = 49 \)
- 49 is divisible by 7:
- \( 49 \div 7 = 7 \)
- Finally, 7 is a prime number.

So, the prime factorization of 11025 is:
\[ 11025 = 3^2 \times 5^2 \times 7^2 \]

**Step 2:** Pair the prime factors.
- We can pair the factors as follows:
- \( (3 \times 3) \), \( (5 \times 5) \), \( (7 \times 7) \)

**Step 3:** Take one factor from each pair.
- The square root of 11025 is:
\[ \sqrt{11025} = 3 \times 5 \times 7 = 15 \times 7 = 105 \]

### 11. Finding the square root of 15876

**Step 1:** Prime factorization of 15876.
- 15876 is divisible by 2:
- \( 15876 \div 2 = 7938 \)
- 7938 is divisible by 2:
- \( 7938 \div 2 = 3969 \)
- 3969 is divisible by 3:
- \( 3969 \div 3 = 1323 \)
- 1323 is divisible by 3:
- \( 1323 \div 3 = 441 \)
- 441 is divisible by 3:
- \( 441 \div 3 = 147 \)
- 147 is divisible by 3:
- \( 147 \div 3 = 49 \)
- 49 is divisible by 7:
- \( 49 \div 7 = 7 \)
- Finally, 7 is a prime number.

So, the prime factorization of 15876 is:
\[ 15876 = 2^2 \times 3^4 \times 7^2 \]

**Step 2:** Pair the prime factors.
- We can pair the factors as follows:
- \( (2 \times 2) \), \( (3 \times 3) \), \( (3 \times 3) \), \( (7 \times 7) \)

**Step 3:** Take one factor from each pair.
- The square root of 15876 is:
\[ \sqrt{15876} = 2 \times 3^2 \times 7 = 2 \times 9 \times 7 = 126 \]

### 12. Finding the square root of 17424

**Step 1:** Prime factorization of 17424.
- 17424 is divisible by 2:
- \( 17424 \div 2 = 8712 \)
- 8712 is divisible by 2:
- \( 8712 \div 2 = 4356 \)
- 4356 is divisible by 2:
- \( 4356 \div 2 = 2178 \)
- 2178 is divisible by 2:
- \( 2178 \div 2 = 1089 \)
- 1089 is divisible by 3:
- \( 1089 \div 3 = 363 \)
- 363 is divisible by 3:
- \( 363 \div 3 = 121 \)
- 121 is divisible by 11:
- \( 121 \div 11 = 11 \)
- Finally, 11 is a prime number.

So, the prime factorization of 17424 is:
\[ 17424 = 2^4 \times 3^2 \times 11^2 \]

**Step 2:** Pair the prime factors.
- We can pair the factors as follows:
- \( (2 \times 2) \), \( (2 \times 2) \), \( (3 \times 3) \), \( (11 \times 11) \)

**Step 3:** Take one factor from each pair.
- The square root of 17424 is:
\[ \sqrt{17424} = 2^2 \times 3 \times 11 = 4 \times 3 \times 11 = 132 \]

### Summary of Results
- \( \sqrt{225} = 15 \)
- \( \sqrt{441} = 21 \)
- \( \sqrt{729} = 27 \)
- \( \sqrt{1296} = 36 \)
- \( \sqrt{2025} = 45 \)
- \( \sqrt{4096} = 64 \)
- \( \sqrt{7056} = 84 \)
- \( \sqrt{8100} = 90 \)
- \( \sqrt{9216} = 96 \)
- \( \sqrt{11025} = 105 \)
- \( \sqrt{15876} = 126 \)
- \( \sqrt{17424} = 132 \)