1. Properties of Exponents
GSE Honors Algebra II
Keeper 20

2. What are rational exponents?
*Exponents that are FRACTIONS. We can rewrite any radical expression with rational exponents!

3. rewriting radical expressions
The rule for rewriting radical expressions with rational exponents is as follows... or

4. Special Cases:

5. Example 1: Rewrite using rational exponents.
5 𝑥

6. Example 2: Rewrite using rational exponents.
7 2 3

7. Example 3: Rewrite using rational exponents.
4 2𝑥 3

8. Example 4: Rewrite using rational exponents.
5 6 𝑥 7

9. Example 5: Rewrite using rational exponents.
4 𝑎 3 𝑏 2 𝑐

10. Example 6: Rewrite in radical form.
𝑥 1 4

11. Example 7: Rewrite in radical form.
5 3 4

12. Example 8: Rewrite in radical form.
3𝑎 5 2

13. Example 9: Rewrite in radical form and simplify completely.
8 5 3

14. Example 10: Rewrite in radical form and simplify completely.
𝑥 3 2

16. What does it mean to simplify?
* Apply the property(s) of exponents.
* Rewrite rational exponents as radicals and simplify if possible.
* We can NEVER leave negative exponents or rational exponents/radicals in the denominator!

17. Helpful Hints:
* Negative exponents have to move to the "opposite" side of the fraction to become positive. * If you end up with a rational exponent in the denominator, rewrite in radical form and then rationalize the denominator.

18. Example 1: Simplify the expression completely.
𝑥 ∙ 𝑥 1 3
Note: If the exponent is a whole number...STOP, that is as far as that piece will go!

19. Example 2: Simplify the expression completely.
16 𝑥

20. Example 3: Simplify the expression completely.
3 𝑥 4 ∙6 𝑥 1 8

21. Example 4: Simplify the expression completely.
𝑥 5 𝑦 −

22. Example 5: Simplify the expression completely.
27 𝑥 12 𝑦

23. Example 6: Simplify the expression completely.
4 𝑧 𝑧 2

24. Example 7: Simplify the expression completely.
2 𝑥 − 1 4 ∙2 𝑦 𝑥 𝑦 − 1 2