1. Properties and Rules for Exponents Properties and Rules for Radicals
Exponents and Radicals
Properties and Rules for Exponents
Product
Rule
Quotient
Power
Zero Exponent
Negative
Exponent
Power of a
quotient
Properties and Rules for Radicals
Principal
square root of a
Negative
Cube root of a
nth root of a
nth root of an
if n is an even and positive integer
if n is an odd and positive integer
Product Rule
for Radicals
Quotient Rule
Like radicals
Conjugates
a ≥0
Bases are the same
Connections
a ≥0
m is the power or exponent
Bases are different

2. Properties and Rules for Radicals
Product Rule
for Radicals
Quotient Rule
Like radicals
Conjugates
We can only multiply/divide radicals with the same root/index.
index or root
radicand
Radicals with the same radicand and index/root. We can only add/subtract like radicals.

3. Simplifying Radicals
Perfect Squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169
Perfect Cubes: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000
Perfect 4ths: 1, 16, 81, 256, 625
Perfect 5ths: 1, 32, 243, 1024
Definition: Numbers whose roots are whole numbers. Look for perfect powers when trying to simplify roots.
Simplify: 9 2 8 5 81 2

4. Simplify each radical Hints:
When there are variables and numbers in the problem, simplify separately
If the root divides evenly into the power of the variable, it is a perfect root. You can take out how many of the variable divide into the root. Whatever is left over stays under the radical.
“For every ______, bring out 1” 9
y·y·y·y·y
8·3
a9; b3b
16·2
z4; z3

5. Multiplying Radicals Rules to follow:
Section 7.3 and 7.4
Rules to follow:
To multiply radicals, the index must be the same. Multiply the values inside and outside the radical separately. If possible, simplify the final answer.
Use the distributive property

6. Dividing Radicals Rules to follow:
To divide radicals, the root must be the same. Use the quotient property and write under a single radical. Simplify the fraction (divide). If possible, simplify the final answer.
Write under a single radical and simplify
27•2 x6
3•5
Write under a single radical and simplify

7. You try: Multiply or divide

8. Adding and Subtracting Radicals
Rules to follow: To add/subtract radicals, they must be like radicals (same root and radicand). Simplify if possible to make radicands the same. Combine ONLY the values outside the radical, the radicand does not change.
Is there a hint about what the common radicand is?
8•3
64•3
4•4
2-16+1

9. You try: Add or Subtract

10. You try: Add or Subtract
For this, you must find a common radicand AND a common denominator.
Find a common radicand first by simplifying the radical

11. Extensions