Math 10: Simplifying Radicals

Don't worry, this lesson won't be that hard. The next lesson will. c:

This lesson we'll cover the rules and methods for simplifying radicals.
And by simplifying, I mean just making the radicals look different by using smaller numbers.

For example, the simplified version of √72 is 6√2.

1. Product Rule for Square Roots and Cube Roots
The product rule is this:

Just get rid of the n when working with square roots and replace the n with 3 for cube roots.

Now, find a factor of that number that is a perfect square.
For example, 4 is a factor of 72 AND it is a perfect square.
So, this is how you would simplify 72:

Notice that the numbers with the red circles are perfect squares.

But 72 is not only divisible by 4 and 18, but also by 2 and 36. (And yeah, 6 and 12 and 8 and 9, but let me teach.)
So here's another way of simplifying:

Notice that this way is much faster. It all depends on the number, its factors, and how you use them.
Also, some numbers can't be simplified, like √2, but say you had a question with two radicals, like √2 X √14. You can then multiply them together, instead of trying to simplify them on their own.

Similarly, for cube roots, you need to find perfect cubes in that number's list of factors.
Just to remind you, cube roots are:

2. Entire and Mixed Radicals

A mixed radical is a number with a number on the outside of a radical sign AND inside of a radical sign.
   EX) 6√2
An entire radical is when there is only one number and it's inside a radical sign.
   EX) √72

As you know, √72 and 6√2 are equal, just two different versions.

3. Un-simplifying Radicals
Multiply the OUTSIDE number depending on whether or not the radical is a cube or a square, and add to the number on the inside.