1. Unit 1, Lesson 8 Rational Exponents and Radicals: Rationalizing the denominator

2. Rational Denominator Rational denominators simply means the bottom number does not have a radical in it. For example, has a rational denominator while doesn’t Which of the following do not have rational denominators: Only has a rational denominator

3. How to rationalize denominators There are two basic concepts to rationalize denominators: • The radicand (the number under the radical sign) is raised to a power that equals the index (the root) • You have to multiply the numerator by the same value as you multiply the denominator (continue on the next slide)

4. How to rationalize denominators Example:  denominator isn’t rational. The radicand, which is 3, is raised to the power of 1and the index (the root) is 2 (remember that if no index is written it is always 2) Using the first concept, you know the radicand has to be raised to the power of 2 to match the index This is accomplished by multiplying by because ∙ = (do you know why?) Now that the power and the index are the same, they will cancel each other out (continued on next slide)

5. How to rationalize denominators Using the second concept, if you multiply the denominator by , you have to multiply the numerator by also (because is the same as multiplying by 1. Doing this doesn’t change the value, it just changes how it looks) So all together it works like this: ∙ = =  you can also think of it as making the denominator a perfect square, in this case 9, which you know has a square root of 3

6. More Examples of Rationalizing Denominators Ex. 1 The index again is 2 so the power needs to be 2 Multiply by to make it or Next, you have to multiply the denom & the num by the same thing, so: ∙ =

7. Another Example Ex. 2  what is the index and what is the power? To figure this out, it might help to think of the rational exponent that corresponds to The index is 2 and the power is 1 ( = 111/2) So, multiply both num & denom by ∙ = = Look closely at the next example; it looks different but it is the same concepts

8. More Examples of Rationalizing Denominators Ex. 3 What do you notice is different from the previous example? In this example, the index is 3, not 2, so the radicand has to be raised to the power of 3 Instead of multiplying by, you have to multiply by ∙ =  remember you add exponents when multiplying with the same base Now, multiply the num by the same thing as the denom: ∙ = =

9. More Examples of Rationalizing Denominators With an Index Larger Than 1 Ex. 4  Index is 4, power is 1 ∙ = = = The 216 comes from cubing 6; and the 3 comes from reducing to

10. Last Example: Radical in the Numerator and the Denominator Ex. 5  keep in mind we only care about a rational denominator; the numerator can have a radical in it ∙ = Ex. 6 ∙ =