Answer:
5. 12√3
6. 46 - 14√5
Explanation:
5.
We want to simplify these radicals. Look at the first term: -3√3. Well, since 3 is a prime number, we cannot square root any of the factors of 3 further, so this is already in its simplest form.
Look at the next term: 3√12. Remember that 12 = 3 * 4, and 4 = 2², so we can actually write 3√12 as 3 * √3 * √4 = 3 * √3 * 2 = 6√3.
Look at the final term: 3√27. Remember that 27 = 3 * 9 and 9 = 3², so we can write 3√27 as 3 * √3 * √9 = 3 * √3 * 3 = 9√3.
Our new expression is -3√3 + 6√3 + 9√3. We can now combine them because they're like terms: 12√3.
6.
Let's use FOIL to expand this. FOIL is first, outer, inner, and last.
-3 and -2 are the first terms, so multiply them: (-3) * (-2) = 6.
-3 and 2√5 are the outer terms, so multiply them: (-3) * (2√5) = -6√5.
4√5 and -2 are the inner terms, so multiply them: (4√5) * (-2) = -8√5.
4√5 and 2√5 are the last terms, so multiply them: (4√5) * (2√5) = 8 * √25 = 8 * 5 = 40.
Now, add all these up:
6 + (-6√5) + (-8√5) + 40 = 46 - 14√5.