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When simplifying radical expressions, one of the most important rules to remember is the property of radicals. This property states that \(\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}\). This means we can combine the numerator and denominator under a single radical.

For example, in the problem \(\frac{\sqrt{200}}{\sqrt{2}}\), we use the property of radicals to rewrite it as \(\sqrt{\frac{200}{2}}\).

This consolidation simplifies the process and makes further simplification easier. Remember, this property only works when the radicals have the same type of root, like both being square roots.

This technique is helpful for breaking down complex expressions into simpler forms that are easier to manage and solve.

Another key concept in this problem is simplifying fractions. Once we combine the radicals using the property of radicals, we need to simplify the fraction inside the radical.

For instance, after transforming \(\frac{\sqrt{200}}{\sqrt{2}}\) into \(\frac{200}{2}\), we simplify it to 100.

Simplifying fractions involves finding the largest common factor between the numerator and the denominator, and dividing both by it. In this case, the largest common factor is 2, which simplifies \(\frac{200}{2}\) down to 100.

By simplifying the fraction inside the radical, we make the final steps of solving the problem much easier.

The final step in our problem focuses on square roots. After simplifying the fraction to 100, we take the square root of this value.

The square root of a number is a value that, when multiplied by itself, gives the original number. For example, \(\sqrt{100} = 10\) because \(10 \times 10 = 100\).

Recognizing perfect squares helps in solving these problems. Perfect squares are numbers like 1, 4, 9, 16, 25, and so on, whose square roots are whole numbers.

Understanding this concept helps us quickly identify \(\root{100}\) as 10. This leads us to the simplified result of our original expression \(\frac{\sqrt{200}}{\sqrt{2}} = 10\).

Using these concepts together—property of radicals, simplifying fractions, and understanding square roots—allows us to break down and solve radical expressions smoothly.