Algebra II Chapter 6: Rational Exponents and Radical Functions Chapter 6 Lesson 1 Evaluate nth Roots and Use Rational Exponents Vocabulary 1
Example 1: Find nth Roots Note: and Example 2: Evaluate Expressions with Rational Exponents Ex: You will not get negative results when evaluating even roots! 2
Example 3: Approximate Roots with Calculator Example 4: Solve Equations Using nth Roots Consider the number and type of answers you might get! Make sure to get the as isolated as possible using subtraction, addition, multiplication, and division properties of equality! 3
Example 5: Use nth Roots in Problem Solving Substitute and solve. Chapter 6 Lesson 2 Apply Properties of Rational Exponents Vocabulary 4
Example 1: Use Properties of Exponents Property For all nonzero real numbers x and y and integers m and n Algebra Numbers Requirement Product of Powers To multiply powers with the same base, add the exponents. Like Bases Quotient of Powers To divide powers with the same base, subtract the exponents. Like Bases Power of a Power To raise one power to another, multiply the exponents. Power of a Product To find the power of a product, apply the exponent to each factor. Like Exponents Power of a Quotient To find the power of a quotient, apply the exponent to the numerator and the denominator. Negative Exponent A nonzero base raised to the negative exponent is equal to the reciprocal of the base raised to the positive exponent. Zero Exponent A nonzero quantity raised to the zero power is equal to 1. Example 1: Use Properties of Exponents State the properties of exponents you used! 5
Example 2: Apply Properties of Exponents Example 3: Use Properties of Radicals State the properties of radicals you used! 6
Example 4: Write Radicals in Simplest Form A radical with index n is in simplest form when: 1. the radicand has no perfect nth powers as factors 2. any denominator has been rationalized Example 5: Add and Subtract Like Radicals and Roots Try to make "like radicals" or "like exponents" by reducing first, then add. 7
Example 6: Simplify Expressions Involving Variables Example 7: Write Variable Expressions in Simplest Form A radical with index n is in simplest form when: 1. the radicand has no perfect nth powers as factors 2. any denominator has been rationalized 8
Example 8: Add and Subtract Expressions Involving Variables Try to make "like radicals" or "like exponents" by reducing first, then add. Chapter 6 Lesson 3 Perform Function Operations and Compositions Vocabulary The domain of the result is the combination of the domain of each, but you must remember to check for values that would make a statement undefined or impossible! For example: 2 could not be in the domain here! Negatives could not be in the domain of! 9
Example 1: Add and Subtract Functions SUPER IMPORTANT: Show the substitution!!! Consider what values that x could possibly be. Reminders: Dividing by zero would result in an undefined solution. The square root of a negative number results in an imaginary number. Example 2: Multiply and Divide Functions SUPER IMPORTANT: Show the substitution!!! Consider what values that x could possibly be. Reminders: Dividing by zero would result in an undefined solution. The square root of a negative number results in an imaginary number. 10
Example 3: Solve a Multi Step Problem Example 4: Standardized Test Practice SUPER IMPORTANT: Show the substitution!!! Use PEMDAS! 11
Example 5: Find Composition Functions SUPER IMPORTANT: Show the substitution!!! When finding domain consider what values that x could possibly be. Reminders: Dividing by zero would result in an undefined solution. The square root of a negative number results in an imaginary number. Example 6: Solve a Multi Step Problem 12
Chapter 6 Lesson 4 Use Inverse Functions Vocabulary Example 1: Find an Inverse Relation Finding the Inverse: 1. Write the original equation in terms of x and y. 2. Switch x and y. 3. Solve for y. 13
Example 2: Verify that Functions are Inverses Show that and. Example 3: Solve a Multi Step Problem Note: In real world models, do not switch the variables. Simply solve for the other one. 14
Example 6: Find the Inverse of a Power Model Note: In real world models, do not switch the variables. Simply solve for the other one. Example 7: Use an Inverse of a Power Model to Make a Prediction 15
Example 5: Find the Inverse of a Cubic Function Make a table of values!!! Notice the reflection about y=x. Example 4: Find the Inverse of a Power Function Make a table of values!!! Notice the reflection about y=x. 16
Chapter 6 Lesson 5 Graph Square Root and Cube Root Functions Vocabulary Reminder: Parent Function Example 1: Graph a Square Root Function State the domain and range, and compare it to the parent function. 17
Example 4: Graph a Translated Square Root Function Example 2: Graph a Cube Root Function State the domain and range, and compare it to the parent function. 18
Example 5: Graph a Translated Cube Root Function Example 3: Solve a Multi Step Problem 19
Chapter 6 Lesson 6 Solve Radical Equations Vocabulary Reminder: Extraneous Solution Example 1: Solve a Radical Equation 20
Example 2: Solve a Radical Equation Given a Function Example 3: Standardized Test Practice 21
Example 4: Solve an Equation with a Radical Exponent Reminder: If you take an even nth root of both sides, your answer could be positive or negative. You should also check for extraneous solutions as well. Example 5: Solve an Equation with an Extraneous Solution 22
Example 6: Solve an Equation with Two Radicals 23