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When we simplify radicals, such as square roots, our goal is to break down the expression into its simplest form. This often involves identifying perfect square factors and reducing the radical accordingly.

Take the example of \(\sqrt{4}\), which we know is 2 because 2 * 2 equals 4. The process here is straightforward because 4 is a perfect square. But not all numbers under the radical will be perfect squares, which is where our example involving \(\sqrt{5}\) comes into play. Since 5 has no square factors other than 1, it remains under the square root. Simplifying radicals makes expressions easier to work with and understand, and can also help when performing more complex algebraic operations.

Rationalizing the denominator means to eliminate any radical present in the denominator of a fraction. This is done to achieve a standard form where no radical is left in the denominator. It usually involves multiplying the numerator and denominator by a number that will cancel out the radical in the denominator.

For instance, with the fraction \(\frac{2}{\sqrt{5}}\), we multiply both the top and bottom by \(\sqrt{5}\) resulting in \(\frac{2\sqrt{5}}{5}\). Notice that \(\sqrt{5} \times \sqrt{5} = 5\), which means the denominator no longer holds a square root, successfully rationalizing it. Rationalizing is particularly useful in further calculations and when comparing the sizes of two quantities.

Dealing with the square root of fractions involves breaking the square root across the numerator and the denominator separately. This concept allows us to tackle each part of the fraction one at a time, making it easier to simplify.

In our exercise, we split \(\sqrt{\frac{4}{5}}\) into \(\frac{\sqrt{4}}{\sqrt{5}}\). Once separated, we simplify the square root of the numerator, which in this case is a perfect square and can be simplified to 2. The square root of the denominator is simplified as much as possible, although in this example it remains as \(\sqrt{5}\) because 5 is prime. Dividing these simplified forms gives us the fraction \(\frac{2}{\sqrt{5}}\), which we can then process further by rationalizing the denominator, if necessary.