Find the square roots of the numbers by expressing each number as the product of primes :
196

Answer

To find the square root of the number 196 by expressing it as the product of its prime factors, we can follow these steps:

### Step 1: Prime Factorization of 196
We start by finding the prime factors of 196. Since 196 is an even number, we can divide it by 2.

- **Divide 196 by 2:**
\( 196 \div 2 = 98 \)
So, we have \( 196 = 2 \times 98 \).

Next, we continue factoring 98.

- **Divide 98 by 2:**
\( 98 \div 2 = 49 \)
Now we have \( 196 = 2 \times 2 \times 49 \).

Now, we factor 49.

- **Divide 49 by 7:**
\( 49 \div 7 = 7 \)
Thus, \( 49 = 7 \times 7 \).

Putting it all together, we have:
\[ 196 = 2 \times 2 \times 7 \times 7 \]

### Step 2: Expressing 196 as a Product of Primes
Now we can express 196 as:
\[ 196 = 2^2 \times 7^2 \]

### Step 3: Finding the Square Root
To find the square root of 196, we take the square root of the prime factorization:
\[ \sqrt{196} = \sqrt{2^2 \times 7^2} \]

Using the property of square roots:
\[ \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \]
we can simplify:
\[ \sqrt{196} = \sqrt{2^2} \times \sqrt{7^2} \]

Calculating each square root:
\[ \sqrt{2^2} = 2 \]
\[ \sqrt{7^2} = 7 \]

Now, multiplying these results together:
\[ \sqrt{196} = 2 \times 7 = 14 \]

### Final Answer
Thus, the square root of 196 is:
\[ \sqrt{196} = 14 \]

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