Simplify √392.

To simplify the square root of 392, we first need to factor the number into its prime components.

Step 1: Factor 392

We can break down 392 as follows:

• 392 is even, so divide by 2: 392 ÷ 2 = 196
• 196 is also even: 196 ÷ 2 = 98
• 98 is even as well: 98 ÷ 2 = 49
• 49 can be factored into: 7 × 7

Step 2: Write the prime factorization

The complete factorization of 392 is:

2 × 2 × 2 × 7 × 7 or 2³ × 7²

Step 3: Simplify the square root

Now, we can simplify the square root:

√392 = √(2³ × 7²)

Using the property of square roots, we can separate the factors:

√392 = √(2² × 2 × 7²) = √(2²) × √(7²) × √2

Final Result

Calculating the square roots gives us:

2 × 7 × √2 = 14√2

Thus, the simplified form of √392 is 14√2.