Dividing Radicals

1. Dividing Radicals

2. Rationalizing Denominators To Rationalize a Denominator Multiply both the numerator and the denominator of the fraction by a radical that will result in the radicand in the denominator becoming a perfect power. Examples: Cannot be simplified further.

3. The conjugate of The conjugate of Conjugates When the denominator of a rational expression is a binomial that contains a radical, the denominator is rationalized. This is done by using the conjugateof the denominator. The conjugate of a binomial is a binomial having the same two terms with the sign of the second term changed.

4. Simplifying Radicals Simplify by rationalizing the denominator:

5. Simplifying Radicals A Radical Expression is Simplified When the Following Are All True • No perfect powers are factors of the radicand and all exponents in the radicand are less than the index. • No radicand contains a fraction. • No denominator contains a radical.

6. Simplifying Radicals Simplify:

7. § 7.6 Solving Radical Equations

8. Radical Equations Aradical equationis an equation that contains a variable in a radicand. To solve radical equations such as these, both sides of the equation are squared.

9. Extraneous Roots In the previous example, anextraneous rootwas obtained when both sides were squared. An extraneous root is not a solution to the original equation. Always check all of your solutions into the original equation. Check: y = 7 Check: y = 0  FALSE!

10. Two Square Root Terms To solve equations with two square root terms, rewrite the equation, if necessary so that there is only one term containing a square root on each side of the equation. Solve the equation: Check: c = 7 

11. Nonradical Terms Solve the equation: Check: b = 84 Not a solution. b = 4 

12. Summary To Solve Radical Equations • Rewrite the equation so that one radical containing a variable is isolated on one side of the equation. • Raise each side of the equation to a power equal to the index of the radical. • Combine like terms. • If the equation still contains a term with a variable in a radicand, repeat steps 1 through 3. • Solve the resulting equation for the variable. • Check all solutions in the original equation for extraneous solutions.

13. Complex Numbers

14. Every imaginary number has as a factor. The , called the imaginaryunit, is denoted by the letter i. Complex Numbers using i An imaginary numberis a number such as It is called imaginary because when imaginary numbers were first introduced, many mathematicians refused to believe they existed!

15. Complex Numbers using i Every number of the form a + bi where a and b are real numbers is a complexnumber. A complex number has two parts: a real part, a, and an imaginary part, b. a + bi

16. Adding and Subtracting To Add or Subtract Complex Numbers • Change all imaginary numbers to bi form. • Add (or subtract) the real parts of the complex numbers. • Add (or subtract) the imaginary parts of the complex numbers. • Write the answer in the form a + bi.

17. Adding and Subtracting Examples:

18. Multiplying To Multiply Complex Numbers • Change all imaginary numbers to bi form. • Multiply the complex numbers as you would multiply polynomials. • Substitute –1 for each i2. • Combine the real parts and the imaginary parts. Write the answer in a + bi form.

19. Multiplying Examples:

20. CAUTION!

21. Dividing To Divide Complex Numbers • Change all imaginary numbers to bi form. • Rationalize the denominator by multiplying both the numerator and the denominator by the conjugate of the denominator. • Substitute –1 for each i2.

22. Dividing Examples:

23. Finding Powers of i The successive powers of i rotate through the four values of i, -1, -i, and 1. in = i if n = 1, 5, 9, … in = 1 if n = 4, 8, 12, … in = -1 if n = 2, 6, 10, … in = -i if n = 3, 7, 11, …