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When simplifying expressions involving square roots, it is important to have a rational (or non-radical) number in the denominator. This is called rationalizing the denominator.

To start, consider an expression like \(\frac{3}{\root{2}}\). This expression has a square root in the denominator, which often makes it less understandable and harder to use in further calculations. To rationalize the denominator, we multiply both the numerator and the denominator by a number that will eliminate the square root in the denominator.

In this case, we use \(\root{2} \). Multiplying by \(\frac{\root{2}} {\root{2}} \) gives us:

\( \frac{3}{\root{2}} \times \frac{\root{2}}{\root{2}} \). This changes the denominator from \(\root{2}\) to 2 (since \(\root{2} \times \root{2} = 2\)).
In short, the process of rationalizing the denominator makes the expression easier to understand and work with by removing any radicals from the denominator.

Multiplying square roots is an important step when simplifying expressions involving radicals.

Let's review \(\frac{3}{\root{2}} \times \root{2}\frac{\root{2}}{\root{2}} \). Here, we multiply the numerators together and the denominators together:

• For the numerator: \((3 \times \root{2})\) results in \(3\root{2}\).


• For the denominator: \(\root{2} \times \root{2} = 2\).

So, multiplying square roots is straightforward but requires careful application to ensure the radicals are handled correctly. Always remember to multiply corresponding parts and simplify as needed.

After multiplying the square roots and rationalizing the denominator, the next step is to simplify the fraction.

Consider the expression we obtained: \( \frac{3 \root{2}}{2} \). In this fraction, we have a radical in the numerator and a rational number in the denominator. Since the denominator is already simplified, our focus is now on the numerator.

The simplified form, \( \frac{3 \root{2}}{2} \), indicates that there is no further simplification needed for the numerator or the denominator.

To check, ensure that the numerator and the denominator have no common factors. This final step assures us that we're left with the simplest form of the expression.