What is the conjugate when dealing with radical expressions?

Here's an example.

Let `a = 2, b = 3, c = 4 and d = 5`.

We will be left with the following:

`2sqrt(3) + 4sqrt(5)`

The conjugate has as goal to make a difference of squares, which is why we use it to rationalize denominators. We can find the conjugate by switching the middle sign in the binomial expression.

Hence, the conjugate of `2sqrt(3) + 4sqrt(5)` is `2sqrt(3) - 4sqrt(5)`.

Let's try multiplying the two expressions to see what happens.

`(2sqrt(3) + 4sqrt(5))(2sqrt(3) - 4sqrt(5)) = 4sqrt(9) + 8sqrt(15) - 8sqrt(15) -16sqrt(25) = 4(3) - 16(5) = 12 - 80 = -68`

So, we start with an expression with lots of irrational numbers, and we multiply it by it's conjugate and get a rational number! Math is so cool sometimes!

Here are a few exercises for your practice. Send me a note when you're ready to be given the answers.

Practice exercises:

`1.` For the following expressions:

•Write the conjugate
•Multiply the expression by its conjugate
•Simplify the expression if necessary

a) `4sqrt(6) + 3sqrt(11)`

b) `sqrt(72) - sqrt(36)`

c) `2sqrt(10) + 3sqrt(16)`

d) `sqrt(19) - sqrt(21)`

Hopefully this helps, and good luck!