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When dealing with square roots in fractions, sometimes the denominator contains an expression that is not simplified. In this case, we need to rationalize the denominator. This involves removing the square root from the denominator to make the fraction easier to handle.

Let's start with the given expression: \( \frac{\sqrt{5}}{\sqrt{2}} \). Here, we aim to eliminate the square root in the denominator. To do this, multiply both the numerator and the denominator by the same square root present in the denominator. In this scenario, we multiply by \( \sqrt{2} \). This ensures that the denominator becomes a rational number since any square root multiplied by itself will result in a whole number.

Multiplying square roots is a straightforward process once you understand that the square root of a product is the product of the square roots. For example, consider the multiplication step in simplifying our expression:

We had \( \frac{\sqrt{5} \times \sqrt{2}}{\sqrt{2} \times \sqrt{2}} \). It's key to note:

• \(\sqrt{5} \times \sqrt{2}\) simplifies to \(\sqrt{10}\), as per the property √(a*b)=√a*√b. This performs the multiplication under a single square root.


• \(\sqrt{2} \times \sqrt{2}\) simplifies to \(\sqrt{4}\).

The point is to combine the terms inside a single square root before simplifying further if possible.

Simplifying radicals means making the expression as simple as possible. This usually involves ensuring there are no square roots in the denominator and combining square roots in the numerator. In our simplified example:

We had \( \frac{\sqrt{10}}{\sqrt{4}} \). Because the square root of 4 is 2, it simplifies to \( \frac{\sqrt{10}}{2} \). Notice:

• \(\sqrt{4}=2\), which makes the fraction more manageable.


• In the expression \( \frac{\sqrt{10}}{2} \), both the numerator and the denominator are simplified to their lowest terms.

By carefully simplifying step-by-step, we've simplified the given expression to remove the complex denominator and combined terms whenever possible.