1. Warm Up
1.) List all of the perfect squares up to The first three are done below.
1, 4, 9,…
Using a calculator, evaluate which expressions are equivalent.
2.) 32
3.) 27
4.) 98
5.) 40
16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225
A. 3 3
B. 4 2
C.2 10
D. 7 2

2. Rational Exponents and Simplest Radical Form
Mr. Riddle

3. Simplest Radical Form =𝟓 𝟑
Expressing in simplest radical form just means simplifying a radical so that there are no more square roots, or cube roots left to find.
Example: 75
= look for perfect squares that factor out
=𝟓 𝟑
Evaluate both and with your calculator.
75 =5 3 ≈8.66

4. Example 1
Simplify by putting the radical expression in simplest radical form.
Step 1: Factor out a perfect square. 
Step 2: Simplify the perfect square. 
Step 3: Are there any more perfect squares in the radicand?
ANSWER:
NO!

5. YOU TRY!!! Simplify the radical expression.
Step 1: Factor out a perfect square. 
Step 2: Simplify the perfect square. 
Step 3: Are there any more perfect squares in the radicand?
ANSWER:
NO!

6. Example 2: Simplest Radical Form with Variables
Simplify the radical expressions.
a.) b.) c.)
When you simplify radical expressions with variables you must make sure that the exponent is an even number!
Example:

7. Example 2: Continued YOU TRY! 𝟐 𝒙 𝟐 𝒚 𝟔𝒚
Simplify the radical expressions!
1.) 24 𝑥 4 𝑦 )
2.) )
YOU TRY!
𝟐 𝒙 𝟐 𝒚 𝟔𝒚

8. Integer Exponents vs. Rational Exponents
Remember that integers are all positive and negative whole numbers.
Example of Integer Exponents:
− 𝑥 9
Rational exponents are written as fractions for exponents. 𝑥

9. Rational Exponents Anatomy

10. Rational Exponents: 𝑏 1 𝑛
For any nonnegative real number, 𝑏 1 𝑛 = 𝑛 𝑏
Examples:
= 25
25 =𝟓
= 3 64
3 64 =𝟒

11. Example 1: Radical and Exponential Forms
Write each expression in radical form, or write each radical in exponential form. A B. 18 You Try! C. (5 𝑥) 1 2 D. 8𝑝

12. Example 2: nth roots
Write each expression in radical form, or each radical in exponential form. A B You Try! C D

13. Rational Exponents: 𝑏 𝑚 𝑛
For any positive real number b and any integers m and n:
𝑏 𝑚 𝑛 = 𝑛 𝑏 𝑚 𝑜𝑟 𝑛 𝑏 𝑚
Examples: =
= 3 64
3 64 =𝟒 = = =𝟒

14. Example 3: Evaluate 𝑏 𝑚 𝑛 expressions.
Simplify. A B C D