1. 10.1 Radical Expressions and Graphs
is the positive square root of a, and is the negative square root of a because
If a is a positive number that is not a perfect square then the square root of a is irrational.
If a is a negative number then square root of a is not a real number.
For any real number a:

2. 10.1 Radical Expressions and Graphs
The nth root of a: is the nth root of a. It is a number whose nth power equals a, so:
n is the index or order of the radical
Example:

3. 10.1 Radical Expressions and Graphs
The nth root of nth powers:
If n is even, then
If n is odd, then
The nth root of a negative number:
If n is even, then the nth root is not a real number
If n is odd, then the nth root is negative

4. 10.1 - Graph of a Square Root Function
(0, 0)

5. 10.2 Rational Exponents Definition:
All exponent rules apply to rational exponents.

6. 10.2 Rational Exponents
Tempting but incorrect simplifications:

7. 10.2 Rational Exponents
Examples:

8. 10.3 Simplifying Radical Expressions
Review: Expressions vs. Equations:
Expressions
No equal sign
Simplify (don’t solve)
Cancel factors of the entire top and bottom of a fraction
Equations
Equal sign
Solve (don’t simplify)
Get variable by itself on one side of the equation by multiplying/adding the same thing on both sides

9. 10.3 Simplifying Radical Expressions
Product rule for radicals:
Quotient rule for radicals:

10. 10.3 Simplifying Radical Expressions
Example:

11. 10.3 Simplifying Radical Expressions
Simplified Form of a Radical:
All radicals that can be reduced are reduced:
There are no fractions under the radical.
There are no radicals in the denominator
Exponents under the radical have no common factor with the index of the radical

12. 10.3 Simplifying Radical Expressions
Pythagorean Theorem: In a right triangle, with the hypotenuse of length c and legs of lengths a and b, it follows that c2 = a2 + b2
Pythagorean triples (integer triples that satisfy the Pythagorean theorem): {3, 4, 5}, {5, 12, 13}, {8, 15, 17} c a
90 b

13. 10.3 Simplifying Radical Expressions
Distance Formula: The distance between 2 points (x1, y1) and (x2,y2) is given by the formula (from the Pythagorean theorem):

14. 10.4 Adding and Subtracting Radical Expressions
We can add or subtract radicals using the distributive property.
Example:

15. 10.4 Adding and Subtracting Radical Expressions
Like Radicals (similar to “like terms”) are terms that have multiples of the same root of the same number. Only like radicals can be combined.

16. 10.4 Adding and Subtracting Radical Expressions
Tempting but incorrect simplifications:

17. 10.5 Multiplying and Dividing Radical Expressions
Use FOIL to multiply binomials involving radical expressions
Example:

18. 10.5 Multiplying and Dividing Radical Expressions
Examples of Rationalizing the Denominator:

19. 10.5 Multiplying and Dividing Radical Expressions
Using special product rule with radicals:

20. 10.5 Multiplying and Dividing Radical Expressions
Using special product rule for simplifying a radical expression:

21. 10.6 Solving Equations with Radicals
Squaring property of equality: If both sides of an equation are squared, the original solution(s) of the equation still work – plus you may add some new solutions.
Example:

22. 10.6 Solving Equations with Radicals
Solving an equation with radicals:
Isolate the radical (or at least one of the radicals if there are more than one).
Square both sides
Combine like terms
Repeat steps 1-3 until no radicals are remaining
Solve the equation
Check all solutions with the original equation (some may not work)

23. 10.6 Solving Equations with Radicals
Example: Add 1 to both sides: Square both sides: Subtract 3x + 7: So x = -2 and x = 3, but only x = 3 makes the original equation equal.

24. 10.7 Complex Numbers Definition:
Complex Number: a number of the form a + bi where a and b are real numbers
Adding/subtracting: add (or subtract) the real parts and the imaginary parts
Multiplying: use FOIL

25. 10.7 Complex Numbers
Examples:

26. 10.7 Complex Numbers
Complex Conjugate of a + bi: a – bi multiplying by the conjugate:
The conjugate can be used to do division (similar to rationalizing the denominator)

27. 10.7 Complex Numbers
Dividing by a complex number: