Find the square roots of the numbers by expressing each number as the product of primes :
441

Answer

To find the square root of 441 by expressing it as the product of its prime factors, we will follow these steps:

### Step 1: Prime Factorization of 441
We start by finding the prime factors of 441. We will divide the number by the smallest prime number until we reach 1.

- Divide 441 by 3 (the smallest prime number):
\[ 441 \div 3 = 147 \]
- Divide 147 by 3:
\[ 147 \div 3 = 49 \]
- Divide 49 by 7:
\[ 49 \div 7 = 7 \]
- Divide 7 by 7:
\[ 7 \div 7 = 1 \]

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

### Step 2: Expressing the Prime Factors in Exponential Form
We can express the prime factors in exponential form:
\[ 441 = 3^2 \times 7^2 \]

### Step 3: Finding the Square Root
To find the square root, we take the square root of each prime factor:
\[ \sqrt{441} = \sqrt{3^2 \times 7^2} \]
Using the property of square roots:
\[ \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \]
We can separate the square roots:
\[ \sqrt{441} = \sqrt{3^2} \times \sqrt{7^2} \]
Calculating the square roots:
\[ \sqrt{3^2} = 3 \quad \text{and} \quad \sqrt{7^2} = 7 \]
Therefore:
\[ \sqrt{441} = 3 \times 7 \]

### Step 4: Final Calculation
Now we multiply the results:
\[ 3 \times 7 = 21 \]

Thus, the square root of 441 is:
\[ \sqrt{441} = 21 \]

### Final Answer
The square root of 441 is **21**.

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