Simplify. Show your work.

5. 12√3

6. 46 - 14√5

Step-by-step 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.

step-by-step explanation:

12: 00 p.m. to 7: 00 p.m. is a total of 7 hours.

4: 00 p.m. to 7: 00 p.m. is a total of 3 hours.

this means that from 4: 00 p.m to 7: 00 p.m. this represents 3/7 of what was lost from 12: 00 p.m. to 7: 00 p.m.

after knowing this, after doing 11,600 - 6,800 = 4,800

4,800 represents 3/7 of what was lost for 7 hours.

we then divide 4,800 by 3 which gets 1,600.

because we know that 1,600 represents 1/7 of how much water was lost, we just have to multiply by 7, which gets 11,200.   after adding 11,200 to how much water was left at 7: 00 p.m. which was 6,800. 11,200 + 6,800 = 18,000

this means that the pool originally had 18,000 gallons in the pool.

you can also check your work by doing 18,000 - 4(1600), which becomes           1800 - 6400 which gets you 11,600.