algebra ii chapter 6: radical...
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Alg II Chapter 6 Notes.notebook
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Algebra IIChapter 6:Rational
Exponents and Radical Functions
Vocabulary
Chapter 6 Lesson 1Evaluate nth Roots and Use Rational Exponents
Alg II Chapter 6 Notes.notebook
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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!
Alg II Chapter 6 Notes.notebook
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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!
Alg II Chapter 6 Notes.notebook
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Example 5: Use nth Roots in Problem Solving
Substitute and solve.
Chapter 6 Lesson 2Apply Properties of Rational Exponents
Vocabulary
Alg II Chapter 6 Notes.notebook
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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 ExponentA 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!
Alg II Chapter 6 Notes.notebook
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Example 2: Apply Properties of Exponents
Example 3: Use Properties of Radicals
State the properties of radicals you used!
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Example 4: Write Radicals in Simplest FormA radical with index n is in simplest form when:1. the radicand has no perfect nth powers as factors2. any denominator has been rationalized
Example 5: Add and Subtract Like Radicals and RootsTry to make "like radicals" or "like exponents" by reducing first, then add.
Alg II Chapter 6 Notes.notebook
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Example 6: Simplify Expressions Involving Variables
Example 7: Write Variable Expressions in Simplest FormA radical with index n is in simplest form when:1. the radicand has no perfect nth powers as factors2. any denominator has been rationalized
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Example 8: Add and Subtract Expressions Involving VariablesTry to make "like radicals" or "like exponents" by reducing first, then add.
Chapter 6 Lesson 3Perform Function Operations and CompositionsVocabulary
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 !
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Example 1: Add and Subtract FunctionsSUPER 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 FunctionsSUPER 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.
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Example 3: Solve a MultiStep Problem
Example 4: Standardized Test Practice
SUPER IMPORTANT: Show the substitution!!!
Use PEMDAS!
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Example 5: Find Composition FunctionsSUPER 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 MultiStep Problem
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Chapter 6 Lesson 4Use Inverse Functions
Vocabulary
Example 1: Find an Inverse RelationFinding the Inverse:1. Write the original equation in terms of x and y.2. Switch x and y.3. Solve for y.
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Example 2: Verify that Functions are Inverses
Show that and .
Example 3: Solve a MultiStep ProblemNote:
In realworld models, do not switch the variables. Simply solve for the other one.
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Example 6: Find the Inverse of a Power Model Note:
In realworld 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
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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 FunctionMake a table of values!!! Notice the reflection about y=x.
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Chapter 6 Lesson 5Graph Square Root and Cube Root FunctionsVocabulary
Reminder:
Parent Function
Example 1: Graph a Square Root Function
State the domain and range, and compare it to the parent function.
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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.
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Example 5: Graph a Translated Cube Root Function
Example 3: Solve a MultiStep Problem
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Chapter 6 Lesson 6Solve Radical EquationsVocabulary
Reminder:
Extraneous Solution
Example 1: Solve a Radical Equation
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Example 2: Solve a Radical Equation Given a Function
Example 3: Standardized Test Practice
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Example 4: Solve an Equation with a Radical ExponentReminder: 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
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Example 6: Solve an Equation with Two Radicals