Is 196 a perfect square ? If so, find the number whose square is 196.

Answer

To determine if 196 is a perfect square and to find the number whose square is 196, we can follow these steps:

### Step 1: Factor the Number
First, we need to factor 196 into its prime factors.

- Start by dividing 196 by the smallest prime number, which is 2:
- \( 196 \div 2 = 98 \)
- Divide 98 by 2 again:
- \( 98 \div 2 = 49 \)
- Now, 49 is not divisible by 2, so we move to the next prime number, which is 7:
- \( 49 \div 7 = 7 \)
- Finally, divide 7 by itself:
- \( 7 \div 7 = 1 \)

So, the prime factorization of 196 is:
\[ 196 = 2 \times 2 \times 7 \times 7 \]
or
\[ 196 = 2^2 \times 7^2 \]

### Step 2: Check for Perfect Square
A number is a perfect square if all the prime factors have even powers.

- In our factorization, we have:
- \( 2^2 \) (even power)
- \( 7^2 \) (even power)

Since both powers are even, we can conclude that 196 is indeed a perfect square.

### Step 3: Find the Square Root
To find the number whose square is 196, we take the square root of the prime factorization.

- The square root of \( 196 \) can be calculated as follows:
\[ \sqrt{196} = \sqrt{2^2 \times 7^2} \]
- Using the property of square roots:
\[ \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \]
- Thus:
\[ \sqrt{196} = \sqrt{2^2} \times \sqrt{7^2} = 2 \times 7 = 14 \]

### Conclusion
Therefore, 196 is a perfect square, and the number whose square is 196 is:
\[ \boxed{14} \]

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