1. Section 7.3 Addition, Subtraction, Multiplication & Division with
Radical Expressions
Adding and subtracting radical expressions is like combining like terms.
With exponential terms, like terms have the same base and same exponent.
You only add the coefficients and keep the variable terms the same:
3x2 + 5x2 = 8x2
Likewise, like radical terms have the same radicand and same index.
You cannot add radical expressions that have different radicands or different indices.
Therefore, if terms have different numbers inside the radical sign and these radical expressions cannot be simplified any more, then you cannot combine them. You can, however, use the distributive property to factor out any like terms.
Simplify:

2. Example:

3. Multiplication & Division of Radical Expressions
When multiplying radical expressions, AS LONG AS THEY HAVE THE SAME INDEX, you can just put everything under the radical sign.
Distributive Property Example:
p.484 #47 (assume t is positive)

4. Box Method: Put the first expression on the side of the box,
Example 3 Multiply
Box Method: Put the first expression on the side of the box,
and the 2nd expression on the top of the box, then multiply the rows and columns to do the FOIL method. Lastly, add up the terms inside the box. Combine the terms that are like terms.
The first box entry looks like this:
Assuming x and y are nonnegative,
Now add everything inside the box:
Example 3 Multiply
Notice that when radical expression has two terms, all radicals disappear when you multiply the expression by its conjugate.
Try this one:

5. Put them in fractional exponent form and use the exponent properties.
What if you are multiplying radical expressions that have the same radicand but don’t have the same index?
Put them in fractional exponent form and use the exponent properties.
What if the expressions have different radicands AND different indices?
Put them both in fractional exponent form, then rewrite each exponent to have the LCD both fractions. Now that they have the same index, you can put them both under the same radical sign.
Let’s do p.484 #41 (again, assuming x is positive)
They have the same index, so just put everything under the radical sign.