square root of 106 simplified

Steps to Solve

1. Identify the number to simplify We start with the number 106. We need to find its prime factors to see if it can be simplified.

2. Find the prime factors of 106 To find the prime factors, we can divide 106 by the smallest prime numbers:

• 106 is even, so divide by 2:

$$ 106 \div 2 = 53 $$

Now we have $106 = 2 \times 53$.

3. Check if the remaining factor is prime Next, we need to check if 53 is a prime number. A prime number has no divisors other than 1 and itself.

The prime numbers less than the square root of 53 (which is approximately 7.28) are 2, 3, 5, and 7. Checking these:

• 53 is not divisible by 2 (since it's odd).
• The sum of digits of 53 is 8, which is not divisible by 3.
• 53 does not end in 0 or 5, so it’s not divisible by 5.
• Finally, dividing 53 by 7 gives a non-integer.

Since 53 is not divisible by any prime numbers up to its square root, it is a prime number.

4. Write the simplified form of the square root Since 106 has been factored into $2 \times 53$ and both factors are prime, the square root can't be simplified further.

Thus, the simplified form of the square root of 106 is:

$$ \sqrt{106} $$