91Ó°ÊÓ

Radical expressions involve roots, such as square roots. The main component of a radical expression is the radical sign (√). For example, \(\sqrt{6}\) is a radical expression where 6 is the radicand.
When working with radical expressions, remember that simplifying them often makes calculations easier. Radicals can sometimes be combined or simplified if their radicands have common factors. Generally, a simplified radical does not have a radicand with a factor that is a perfect square (except 1).
Starting with the basics can make dealing with more complex radical expressions less intimidating.

Multiplying radicals follows a simple rule: multiply the radicands together under a single radical. For example, \(\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}\).
In our exercise, we start with \(\frac{\sqrt{6}}{\sqrt{7}} \cdot \frac{\sqrt{14}}{\sqrt{3}}\). To simplify, first multiply the numerators: \(\sqrt{6} \cdot \sqrt{14}\). This gives us \(\sqrt{84}\) because 6 times 14 equals 84.
Next, multiply the denominators: \(\sqrt{7} \cdot \sqrt{3}\). This equals \(\sqrt{21}\), since 7 times 3 equals 21.
Always remember to combine the radicals before further steps like division or additional simplification.

Simplifying fractions, especially those that involve radicals, is a crucial skill. It's easier to work with simpler expressions. Our goal is to convert complicated fractions into simpler forms.
In the example \(\frac{\sqrt{84}}{\sqrt{21}}\), the next step is to combine under a single radical: \(\sqrt{\frac{84}{21}}\). Simplifying the fraction inside the radicand is vital. 84 divided by 21 equals 4. This reduces the expression inside the radical: \(\sqrt{4}\).
Finally, when the radicand itself is a perfect square, take the square root to simplify it fully. The square root of 4 is 2. So, we started with a complex fraction involving radicals and simplified it down to 2.
Following these steps ensures clear, easy-to-understand results and makes working with radical expressions more manageable.