1. Radical Expressions
Part II

2. Rationalizing Radicals
You studied how to multiply with radicals to produce a simplified radical using the product property of radicals: 𝑎𝑏 = 𝑎 ⋅ 𝑏
We can do the same with radical fractions using the quotient property of radicals 𝑎 𝑏 = 𝑎 𝑏
For example,
= ⋅ = ⋅ = ⋅ ⋅ 3 = 2 2 ⋅ 2 5⋅ 3 =

3. Rationalizing Radicals
However, with fractions we often need to take one further step
It is preferable not to have radicals in the denominator
When we have a radical fraction with a radical in the denominator, we will rationalize the denominator
We will do this by using the following property
𝑎 2 = 𝑎 ⋅ 𝑎 = 𝑎⋅𝑎 = 𝑎 2 =𝑎
That is, we can remove a radical by squaring it

4. Rationalizing Radicals
To rationalize a denominator, you will multiply the fraction by 1 by creating a fraction of the denominator over itself
For example, the radical fraction should be rationalized because it has a radical in the denominator
To do this, we multiply by 1=
⋅ = 2⋅ =

5. Rationalizing Radicals
⋅ = 2⋅ =
The effect is to remove the radical from the denominator
Another example
16 5 = = ⋅ = =

6. Guided Practice
Simplify each radical.
35 36
13 28
18 11

7. Guided Practice Simplify each radical. 35 36 = 35 36 = 35 6
= =
= = ⋅7 = ⋅ = 13⋅7 2⋅7 =
= = 2⋅ = ⋅ =

8. Using A Conjugate to Simplify Radicals
Radicals can be added to non-radicals
Since we wish to express exact values, we will not actually combine terms
However, ≈6.414
Any number of this form (a radical added to a radical or two different radicals added together) has a conjugate
For any radical expression of the form 𝑎+ 𝑏 , its conjugate is 𝑎− 𝑏

9. Using Conjugates to Simplify Radicals
What makes conjugates special?
Let’s multiply 𝑎+ 𝑏 𝑎− 𝑏 (copy this in your notes)
𝑎+ 𝑏 𝑎− 𝑏
=𝑎 𝑎+ 𝑏 − 𝑏 𝑎+ 𝑏
= 𝑎 2 +𝑎 𝑏 −𝑎 𝑏 − 𝑏 2
= 𝑎 2 −𝑏
The result of multiplying two conjugates is to eliminate the radical

10. Using Conjugates to Simplify Radicals
Radical fractions may have an expression like 𝑎+ 𝑏 in the denominator
To rationalize the denominator, we multiply by the conjugate over itself
Example
⋅ 7− 2 7− 2 = 3 7− − 2 = 21− −2 = 21−
This, too is a number approximately equal to , but we prefer to leave it in exact form

11. Using Conjugates to Simplify Radicals
Another example
2 5− 6
= − 6 ⋅
= −6 = ≈

12. Guided Practice Use conjugates to rationalize the denominator. 2 1− 3
2 1− 3
−2 4− 8

13. Guided Practice Use conjugates to rationalize the denominator.
2 1− 3 = 2 1− 3 ⋅ = −3 = −2 =− ≈−2.732
= ⋅ 3− 7 3− 7 = 3 3 − −7 = 3 3 − ≈
−2 4− 8

14. Practice 9
Handout