Simplify the following: .

First, we'll start inside the parentheses by finding the cube root of -27. In other words, what number multiplied by itself 3 times gives us -27?

Next, things finish up nicely by squaring -3. Be careful, (-3)² is different than -3².

(-3)² = 9

Simplify the following: .

We can start in the first set of parentheses here. This problem has actually made life fairly easy, since the fifth root of 6⁵ is just going to be 6 again. If you don't believe us, put that calculator to work.

Working in the other parentheses, we can evaluate the square root of 25.

= 6(5)³

Before we multiply, PEMDAS reminds us to take care of that exponent.

(6)(5)³ = 6(125)

Finally, multiply to finish things up. We dare you to do it mentally.

6(125) = 750

Simplify the following: .

Step one is to change into a product of prime numbers. There's more than one way to do this. Our favorite is shown.

To change to simple radical form, we need to pull out one 3 for each pair inside the square root, since .

Last—get ready for it—we multiply.

Simplify the following: .

While this can be solved a number of different ways, we're just going to change all the square roots to simple radical form first. Product of primes, here we come.

Next, we pull out our pairs and multiply.

Finally, we can multiply our first two terms together, keeping what's outside of the square roots outside, and what's inside, inside. 

This actually gives us our final answer, since 12√6 and 4√2 are not like terms. It would be like adding 12x and 4y together. Not happenin'.