Square roots. Woohoo. Want to check some work I did.

1. Perform indicated operations
3sqrt[3]+2sqrt[27]-sqrt[12]
3sqrt[3]+2sqrt[3*9]-sqrt[2*6]
3sqrt[3]+3*2sqrt[3]-2sqrt[3]
3sqrt+6sqrt[3]-2sqrt[3]
= 7sqrt[3]

2.Simplify
sqrt[49x^12y^4z^8]
= 7x^6y^2z^4

3.Multply
(8sqrt[6]+3sqrt[2])(4sqrt[6]-5sqrt[2]
32sqrt[36]-40sqrt[12]+12sqrt[12]-15sqrt[4]
32*6-40sqrt[12]+12sqrt[12]-15*2
192 - 28sqrt[12] -30
162 - 28sqrt[12]

or am i missing a step?
192-40sqrt[4*3]+12sqrt[4*3]-30
162-40*4sqrt[3]+12*4sqrt[3]
162-160sqrt[3]+48sqrt[3]
162- 128sqrt[3] which I think turns into 162 -56sqrt[3]

Let's go through each step together to check your work:

1. Perform indicated operations:
Starting with 3√3 + 2√27 - √12
You correctly simplified the square roots of 27 and 12:
3√3 + 2√(3*9) - √(2*6)
3√3 + 2√3*√9 - √2*√6
3√3 + 2√3*3 - √2*√6
3√3 + 6√3 - 2√6
Combine like terms:
(3+6)√3 - 2√6
9√3 - 2√6

2. Simplify:
Now let's simplify √(49x^12y^4z^8)
The square root of 49 is 7 and the variables inside the square root can be simplified to their respective exponents:
√(7^2x^12y^4z^8)
Now simplify each of the variables:
7x^6y^2z^4

3. Multiply:
Now let's multiply (8√6 + 3√2) by (4√6 - 5√2)
Using the FOIL method, multiply each term:
(8√6 * 4√6) + (8√6 * -5√2) + (3√2 * 4√6) + (3√2 * -5√2)
32√36 - 40√12 + 12√12 - 15√4
Now simplify the square roots:
32*6 - 40√(2*6) + 12√(2*6) - 15*2
192 - 40√12 + 12√12 - 30
Combine like terms:
192 - 28√12 - 30
162 - 28√12

You made an error in your last step. When combining like terms, it should be 162 - 28√12, not 162 - 56√3.

Let's check your work step by step.

1. Perform indicated operations:
You correctly simplified each term under the square root symbols:
3sqrt[3] becomes 3 * sqrt(3),
2sqrt[27] becomes 2 * sqrt(27),
sqrt[12] becomes sqrt(12).

Next, you simplified further:
3 * sqrt(3) + 2 * sqrt(27) - sqrt(12)
= 3 * sqrt(3) + 2 * sqrt(3 * 9) - sqrt(2 * 6)
= 3 * sqrt(3) + 3 * 2 * sqrt(3) - 2 * sqrt(3)
= 3 * sqrt(3) + 6 * sqrt(3) - 2 * sqrt(3)
= (3 + 6 - 2) * sqrt(3)
= 7 * sqrt(3)

So, your answer 7sqrt[3] is correct.

2. Simplify:
You correctly simplified the expression sqrt(49x^12y^4z^8) to 7x^6y^2z^4. Good job!

3. Multiply:
You started by distributing each term from the first expression (8sqrt[6]+3sqrt[2]) to each term in the second expression (4sqrt[6]-5sqrt[2]), which is correct.
Then, you simplified each term under the square root symbols:
32sqrt[36] becomes 32 * sqrt(36),
-40sqrt[12] becomes -40 * sqrt(12),
12sqrt[12] becomes 12 * sqrt(12),
-15sqrt[4] becomes -15 * sqrt(4).

Now, let's simplify further:
32 * sqrt(36) - 40 * sqrt(12) + 12 * sqrt(12) - 15 * sqrt(4)
= 32 * sqrt(6^2) - 40 * sqrt(2 * 6) + 12 * sqrt(2 * 6) - 15 * sqrt(2^2)
= 32 * 6 - 40 * sqrt(2 * 6) + 12 * sqrt(2 * 6) - 15 * 2
= 192 - 40 * sqrt(12) + 12 * sqrt(12) - 30
= 192 - 28 * sqrt(12) - 30

Here's where you missed a step:
To simplify further, we can factor out the square root of 12:
= 192 - 28 * sqrt(4 * 3) - 30
= 192 - 28 * 2 * sqrt(3) - 30
= 192 - 56 * sqrt(3) - 30
= 162 - 56 * sqrt(3)

So, your final answer should be 162 - 56sqrt[3], not 162 - 28sqrt[3].

Overall, good job on most of your work!