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Prime factorization is a technique used to break down a number into its basic building blocks — the prime numbers. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves. To factorize a number, you'll divide by the smallest prime number possible, and continue the process with the quotient until you are left with a prime number.

To simplify a radical, like \(\sqrt{175}\), we begin by finding its prime factors. For 175, start by dividing by the smallest prime, 2. Since 175 is odd, it isn’t divisible by 2. Check the next smallest prime, which is 3; 175 isn't divisible by 3 either. Moving on to 5 because 175 ends in a 5 (indicating it's divisible by 5), we find:

• 175 ÷ 5 = 35
• 35 ÷ 5 = 7

35 is also divisible by 5, and what remains is 7, which is a prime number. Now, we have the prime factorization: \(175 = 5^2 \times 7\). This breakdown is crucial for simplifying radicals.

Understanding the properties of square roots is key to simplifying radical expressions. One essential property is that the square root of a product is the product of the square roots:

\[\sqrt{ab} = \sqrt{a} \times \sqrt{b}\]

This property allows us to split a radical into more manageable pieces. When simplifying \(\sqrt{175}\), we use its factor form, \(5^2 \times 7\), allowing application of this property:

• \(\sqrt{175} = \sqrt{5^2 \times 7}\)
• \(= \sqrt{5^2} \times \sqrt{7}\)

This demonstrates how square root properties simplify radicals by separating factors into individual square roots. Identifying and applying these properties make simplifying radicals a clearer process.

Once a number has been factorized into primes, and properties of square roots have been applied, it's time for simplification. Simplifying a radical means finding the square root of any perfect square within the factorization and reducing it.

Take for example \(\sqrt{175}\), which we broke down to \(\sqrt{5^2} \times \sqrt{7}\). The expression \(\sqrt{5^2}\) simplifies to 5 because 5 squared (\(5^2 = 25\)) is a perfect square, and the square root of 25 is 5:

• \(\sqrt{5^2} = 5\)

For the leftovers within the square root \((\sqrt{7})\), since 7 is not a perfect square, it remains under the root:

• So, \(\sqrt{175} = 5 \times \sqrt{7}\)

Simplifying square roots not only makes expressions easier to manage but also crucial in solving more complex mathematical problems.