6. .

To simplify the expression , we can break down into . Thus, . The expression now becomes .

For the second expression, , start by simplifying as and as . This gives us .

Lastly, for , rewrite as , yielding . Then simplifies to , so we have . Finally, , thus . Bringing it all together results in .

In short, the simplified answers are , , and respectively.

Alright, let’s dig a little more into the world of square roots and their simplifications! Square roots have been utilized for centuries, from Ancient Babylonians who approximated them to the more formalized techniques developed in the Islamic Golden Age by mathematicians like Al-Khwarizmi, who approached these calculations with great rigor. Their curiosity sparked the interest we still maintain in these seemingly simple expressions!

Now, how about we keep an eye out for some common traps people fall into while simplifying? One common mistake is forgetting the coefficients when pulling square roots from under the radical sign. Always remember, can be combined as ! Another pitfall is misidentifying perfect squares, which can lead you astray. So, knowing those perfect squares and keeping track of signs will keep your math robust and your answers shiny!