1. Dear Power point User,
This power point will be best viewed as a slideshow. At the top
of the page click on slideshow, then click from the beginning.

2. Radicals and Rational Exponents
Chapter 10

3. 10 Radicals and Rational Exponents 10.1 Finding Roots
10.3 Simplifying Expressions Containing Square Roots
10.4 Simplifying Expressions Containing Higher Roots
10.5 Adding, Subtracting, and Multiplying Radicals
10.6 Dividing Radicals
Putting it All Together
10.7 Solving Radical Equations
10.8 Complex Numbers

4. 10.8
Complex Numbers
We have seen in previous sections that the square root of a negative number does
not exist in the real number system because there is no real number that, when
squared, will result in a negative number.
For example, is not a real number because there is no real number whose
square is -4.
The square roots of negative numbers do exist, however, under another system of
numbers called complex numbers. Before we define a complex number, we must
Define the number i. the number i is called an imaginary number.
Note

5. The following table lists some examples of complex numbers and their real and
Imaginary parts.
Note
Since all real numbers, a, can be written in the form a + 0i, all real numbers are also
complex numbers.

6. Example 1
Solve.
Solution
Note

7. Multiply and Divide Square Roots Containing Negative Numbers
Note
Example 2
Multiply and Simplify.
Solution
Write each radical in terms of i before multiplying.
Multiply
Replace with -1.

8. Add and Subtract Complex Numbers
Just as we can add, subtract, and multiply, and divide real numbers, we can perform
All of these operations with complex numbers.
Example 3
Add or Subtract.
Solution
Add real parts together and imaginary parts
together.
Distributive property.
Add real parts together and imaginary parts
together.

9. Multiply Complex Numbers
We multiply complex numbers just like we would multiply polynomials There may be
An additional step, however. Remember to replace with -1.
Example 4
Add or Subtract.
Solution
Distributive property.
F O I L
You can use FOIL to multiply.
Replace with -1.
Combine like terms.

10. Multiply a Complex Number by Its Conjugate
The product of a complex number and its conjugate is always a real number.
F O I L
Replace with -1.

11. Example 5
Solution
Example 6
Solution

12. Divide Complex Numbers
Example 7
Solution
Multiply the numerator and denominator by the conjugate
of the denominator.
Multiply numerators and use to
multiply the denominators .
Write quotient in the form