Week 5 – Precalculus – Multiplying Mixed Radical Expressions

Hello! This week I decided to cover multiplying mixed radical expressions. As we learned before, mixed radicals are any radical with a coefficient that is different from 1. They are considered the more simplified version as compared to entire fractions. It is important to note that not all entire radicals can be changed into a mixed radical because the radicand does not have a factor that is a perfect square (besides one).

Something important to remember before I jump into this lesson is that multiplying mixed radicals does not follow the same rules as adding or subtracting mixed radicals. When you are adding and subtracting mixed radicals, you require like terms. When you are multiplying or dividing mixed radicals, like terms are not required. Keep this in mind as I continue to our lesson.

Say you wanted to multiply $\left(5-\sqrt{2}\right)\left(5+\sqrt{2}\right)$ .

If you recall back to earlier lessons in Grade 10, you may remember distributive property. Distributive property is used to solve expressions easily by distributing a number to the numbers given in brackets. For example, if we apply the distributive property of multiplication to a simpler expression, we would solve it in the following way: 6(3 + 6) = (6 x 3) + (6 x 6) = 18 + 36 = 54.

Now let’s apply it distributive property to the expression above.

5 will multiply with 5, and then $\sqrt{2}$ will multiply with 5 (note the invisible 1 next to the radical).

Notice there is another digit in each bracket. This means we must use distributive property a second time.

$-\sqrt{2}$ will multiply with the second $\sqrt{2}$ .

The expression should look like this: $25+5\sqrt{2}-5\sqrt{2}-\sqrt{4}$ .

Take a look at this result and notice how $5\sqrt{2}-5\sqrt{2}$ are the same. When we remember that adding and subtracting mixed radicals requires like terms, we know that these will cancel out each other. This will leave us with 25 $-\sqrt{4}$ .