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

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

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: 9292 8585 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 a 9 ; b 3 b 16·2 z 4 ; z 3

5. When a is nonnegative, n can be any number greater than 1. When a is negative, n can be any odd natural number greater than 1 Rational Exponents

6. For any natural numbers m and n (n ≠ 0) and any real number a for which exists Positive Rational Exponents

7. For any rational number m/n and any nonzero real number a for which Negative Rational Exponents

8. Multiplying Radicals 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

9. 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. 272 x 6 3535 Write under a single radical and simplify

10. You try: Multiply or divide

11. 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? 8383 4 643 2-16+1

12. You try: Add or Subtract

13. For this, you must find a common radicand AND a common denominator. Find a common radicand first by simplifying the radical

14. Extensions