# What is the conjugate when dealing with radical expressions?

Aug 28, 2016

Here's an example.

Let $a = 2 , b = 3 , c = 4 \mathmr{and} d = 5$.

We will be left with the following:

$2 \sqrt{3} + 4 \sqrt{5}$

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 $2 \sqrt{3} + 4 \sqrt{5}$ is $2 \sqrt{3} - 4 \sqrt{5}$.

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

$\left(2 \sqrt{3} + 4 \sqrt{5}\right) \left(2 \sqrt{3} - 4 \sqrt{5}\right) = 4 \sqrt{9} + 8 \sqrt{15} - 8 \sqrt{15} - 16 \sqrt{25} = 4 \left(3\right) - 16 \left(5\right) = 12 - 80 = - 68$

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) $4 \sqrt{6} + 3 \sqrt{11}$

b) $\sqrt{72} - \sqrt{36}$

c) $2 \sqrt{10} + 3 \sqrt{16}$

d) $\sqrt{19} - \sqrt{21}$

Hopefully this helps, and good luck!