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Prime factorization plays a crucial role in simplifying square roots. To factorize a number, you break it down into its prime factors - the prime numbers that multiply together to give that original number. For example, the number 500 can be broken down into its prime factors as follows:

• First, divide by the smallest prime number, 2: 500 ÷ 2 = 250
• Continue dividing by 2: 250 ÷ 2 = 125
• Next, since 125 is not divisible by 2, we move to the next smallest prime number, 5: 125 ÷ 5 = 25
• Divide by 5 again: 25 ÷ 5 = 5
• Finally, 5 is divided by 5: 5 ÷ 5 = 1

So, the prime factorization of 500 is: \[500 = 2^2 \times 5^3\]
This factorization helps us to simplify the square root in the next steps.

A radical expression involves a root, such as a square root. To simplify square roots, you can use the prime factorization method mentioned earlier. Once you have the prime factors, you can pair the prime factors and bring them outside the radical.
For example, let's simplify \( \sqrt{500} \) using the prime factors found earlier: \[ \sqrt{500} = \sqrt{2^2 \times 5^3} \]
Simplify each part by removing the square roots of paired primes:
\[ \sqrt{500} = \sqrt{2^2} \times \sqrt{5^3} \]
Which comes out to: \[ \sqrt{2^2} = 2 \] and \[ \sqrt{5^3} = 5 \times \sqrt{5} \], simplifying further:
\[ \sqrt{500} = 2 \times 5 \times \sqrt{5} = 10 \sqrt{5} \]
Now, the square root is in its simplest form as a radical expression.

Applying a negative sign to a simplified square root expression is straightforward but important. Once you have simplified the expression inside the square root, as in the previous example where \( \sqrt{500} = 10 \sqrt{5} \), you simply place the negative sign in front of this simplified form.
Thus, the final step in the given problem is: \[ - \sqrt{500} = -10 \sqrt{5} \]
Adding a negative sign doesn't change the complexity of the expression, but it's crucial to maintain accuracy.
Always make sure to apply any positive or negative signs appropriately as per the problem requirement.