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

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.

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

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

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

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

You try: Multiply or divide

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

You try: Add or Subtract

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

Extensions