Perfect Squares

= 2 = 4 = 5 = 10 = 12

Radicals Students are expected to: * Demonstrate an understanding of the role of irrational numbers in applications. * Approximate square roots. * Demonstrate an understanding of and apply properties to operations involving square roots. * Apply the Pythagorean Theorem. * Use inductive and deductive reasoning when observing patterns, developing properties, and making conjectures.

What are radicals?  A Radical is the root of a number.

What are radicals?  A Radical is the root of a number.

What are radicals?  A Radical is the root of a number. coefficientradical

What are radicals?  For example:  A Radical is the root of a number. coefficientradical radical sign

What are radicals?  A Radical is the root of a number.  For example: coefficientradical

What are radicals?  A Radical is the root of a number.  For example: 5 is the square root of 25 because 5x5 = 25 coefficientradical

What are radicals?  A Radical is the root of a number.  For example: 5 is the square root of 25 because 5x5 =25 coefficientradical

What are radicals?  A Radical is the root of a number.  For example: 5 is the square root of 25 because 5x5 = 25 2 is the cubed root of 8 because 2x2x2 = 8. coefficientradical

Determining Roots  You can determine the root using mental math.

Determining Roots  You can determine the root using mental math.  Example:

Determining Roots  You can determine the root using mental math.  Example:

Determining Roots  You can determine the root using mental math.  Example:  You can also find the root using your calculator.

Determining Roots  You can determine the root using mental math.  Example:  You can also find the root using your calculator.  Example:

Determining Roots  You can determine the root using mental math.  Example:  You can also find the root using your calculator.  Example:

Determining Roots  You can determine the root using mental math.  Example:  You can also find the root using your calculator.  Example:  Be careful not to hit the wrong button on your calculator!

Let’s investigate!

I. Solve for the hypotenuse of the different triangles in your geoboard paper. II. At the back of your paper, draw a square with an area of: a)34 b)41 c)52 d)29

Simplifying Square Roots

= = = = = = = = = = Perfect Square Factor * Other Factor LEAVE IN RADICAL FORM

= = = = = = = = = = Perfect Square Factor * Other Factor LEAVE IN RADICAL FORM

= = = = = = = = = = Perfect Square Factor * Other Factor LEAVE IN RADICAL FORM

Adding and Subtracting Radicals  Only like radicals can be added or subtracted.  Like radicals can have different coefficients as long as the radicals are the same.  Subtract the coefficients but keep the same radical.  You may have to simplify your question first, to get a like radical.

Adding and Subtracting Radicals  Only like radicals can be added or subtracted.  Like radicals can have different coefficients as long as the radicals are the same.  Subtract the coefficients but keep the same radical.  You may have to simplify your question first, to get a like radical. Example 1:

Adding and Subtracting Radicals  Only like radicals can be added or subtracted.  Like radicals can have different coefficients as long as the radicals are the same.  Subtract the coefficients but keep the same radical.  You may have to simplify your question first, to get a like radical. Example 1:

Adding and Subtracting Radicals  Only like radicals can be added or subtracted.  Like radicals can have different coefficients as long as the radicals are the same.  Subtract the coefficients but keep the same radical.  You may have to simplify your question first, to get a like radical. Example 1: Example 2:

Adding and Subtracting Radicals  Only like radicals can be added or subtracted.  Like radicals can have different coefficients as long as the radicals are the same.  Subtract the coefficients but keep the same radical.  You may have to simplify your question first, to get a like radical. Example 1: Example 2:

Adding and Subtracting Radicals  Only like radicals can be added or subtracted.  Like radicals can have different coefficients as long as the radicals are the same.  Subtract the coefficients but keep the same radical.  You may have to simplify your question first, to get a like radical. Example 1: Example 2: Example 3:

Adding and Subtracting Radicals  Only like radicals can be added or subtracted.  Like radicals can have different coefficients as long as the radicals are the same.  Subtract the coefficients but keep the same radical.  You may have to simplify your question first, to get a like radical. Example 1: Example 2: Example 3:

 One common mistake is adding without having like radicals. Common Mistakes Made While Adding and Subtracting Radicals

 One common mistake is adding without having like radicals. Common Mistakes Made While Adding and Subtracting Radicals

 One common mistake is adding without having like radicals.  Always make sure that the radicals are similar. Common Mistakes Made While Adding and Subtracting Radicals

 One common mistake is adding without having like radicals.  Always make sure that the radicals are similar.  Another common mistake is forgetting to simplify. Common Mistakes Made While Adding and Subtracting Radicals

 One common mistake is adding without having like radicals.  Always make sure that the radicals are similar.  Another common mistake is forgetting to simplify. Common Mistakes Made While Adding and Subtracting Radicals

 One common mistake is adding without having like radicals.  Always make sure that the radicals are similar.  Another common mistake is forgetting to simplify.  This question is not done! The answer is:

Multiplying Radicals Steps Example:

Multiplying Radicals Steps Example:

Multiplying Radicals 1. Multiply coefficients together Steps Example: 1.

Multiplying Radicals 1. Multiply coefficients together 2. Multiply radicals together Steps Example: 1. 2.

Multiplying Radicals 1. Multiply coefficients together 2. Multiply radicals together 3. Simplify radicals if possible Steps Example:

Examples of Multiplying Radicals  Correct example:

Examples of Multiplying Radicals  Correct example:

Examples of Multiplying Radicals  Correct example:

Examples of Multiplying Radicals  Correct example:  Be careful to multiply the coefficients with coefficients and radicals with radicals.

Examples of Multiplying Radicals  Incorrect Example:  Correct example:  Be careful to multiply the coefficients with coefficients and radicals with radicals.

Examples of Multiplying Radicals  Incorrect Example:  Correct example:  Be careful to multiply the coefficients with coefficients and radicals with radicals.

Examples of Multiplying Radicals  Correct example:  Be careful to multiply the coefficients with coefficients and radicals with radicals.  Incorrect Example:  This is Wrong! Don’t multiply these together!

+ To combine radicals: combine the coefficients of like radicals

Simplify each expression

Combined Operations with Radicals  You follow the same steps with these as you do with polynomials.

Combined Operations with Radicals  You follow the same steps with these as you do with polynomials.  Use the distribution property.

Combined Operations with Radicals  You follow the same steps with these as you do with polynomials.  Use the distribution property.  Example:

Combined Operations with Radicals  You follow the same steps with these as you do with polynomials.  Use the distribution property.  Example:

Combined Operations with Radicals  You follow the same steps with these as you do with polynomials.  Use the distribution property.  Example:

Combined Operations with Radicals  You follow the same steps with these as you do with polynomials.  Use the distribution property.  Example:

Common Mistakes made while doing combined operations  Be careful to multiply correctly.

Common Mistakes made while doing combined operations  Be careful to multiply correctly.  Correct Answer:

Common Mistakes made while doing combined operations  Be careful to multiply correctly.  Correct Answer:

Common Mistakes made while doing combined operations  Be careful to multiply correctly.  Correct Answer:  Incorrect Answer:

Common Mistakes made while doing combined operations  Be careful to multiply correctly.  Correct Answer:  Incorrect Answer:

Simplify each expression: Simplify each radical first and then combine.

Lets Play! Simplify each expression.

Homework

Dividing Radicals (part 1)  There are 4 steps to dividing radicals.

Dividing Radicals (part 1)  There are 4 steps to dividing radicals.  1. Reduce coefficients and/or radicals by common factor (if you can).

Dividing Radicals (part 1)  There are 4 steps to dividing radicals.  1. Reduce coefficients and/or radicals by common factor (if you can).  2.Take out any perfect squares (if possible).

Dividing Radicals (part 1)  There are 4 steps to dividing radicals.  1. Reduce coefficients and/or radicals by common factor (if you can).  2.Take out any perfect squares (if possible).  Rationalize denominators

Dividing Radicals (part 1)  There are 4 steps to dividing radicals.  1. Reduce coefficients and/or radicals by common factor (if you can).  2.Take out any perfect squares (if possible).  Rationalize denominators.  Reduce coefficients again.

Dividing Radicals (part 1)  There are 4 steps to dividing radicals.  1. Reduce coefficients and/or radicals by common factor (if you can).  2.Take out any perfect squares (if possible).  3. Rationalize denominators.  4. Reduce coefficients again.  Example:

Dividing Radicals (part 1)  There are 4 steps to dividing radicals.  1. Reduce coefficients and/or radicals by common factor (if you can).  2.Take out any perfect squares (if possible).  Rationalize denominators.  Reduce coefficients again.  Example:

Dividing Radicals (part 1)  There are 4 steps to dividing radicals.  1. Reduce coefficients and/or radicals by common factor (if you can).  2.Take out any perfect squares (if possible).  Rationalize denominators.  Reduce coefficients again.  Example: Divide top and bottom by

Common Errors Done While Dividing Radicals  Be careful not to divide a coefficient by a radical.

Common Errors Done While Dividing Radicals  Be careful not to divide a coefficient by a radical.  Example of this error:

Common Errors Done While Dividing Radicals  Be careful not to divide a coefficient by a radical.  Example of this error:

Common Errors Done While Dividing Radicals  Be careful not to divide a coefficient by a radical.  Example of this error:  The correct answer to this question is:

Common Errors Done While Dividing Radicals  Be careful not to divide a coefficient by a radical.  Example of this error:  The correct answer to this question is:

Common Errors Done While Dividing Radicals  Be careful not to divide a coefficient by a radical.  Example of this error:  The correct answer to this question is: Multiply top and bottom by

Dividing Radicals (part 2)  When rationalizing binomial denominators, multiply both sides by the conjugate.

Dividing Radicals (part 2)  When rationalizing binomial denominators, multiply top and bottom by the conjugate.

Dividing Radicals (part 2)  When rationalizing binomial denominators, multiply top and bottom by the conjugate. Binomial denominator.

Dividing Radicals (part 2)  When rationalizing binomial denominators, multiply top and bottom by the conjugate.  Example: Binomial denominator.

Dividing Radicals (part 2)  When rationalizing binomial denominators, multiply top and bottom by the conjugate.  Example: Binomial denominator.

Dividing Radicals (part 2)  When rationalizing binomial denominators, multiply top and bottom by the conjugate.  Example: Binomial denominator. Multiply numerator and denominator by

Dividing Radicals (part 2)  When rationalizing binomial denominators, multiply top and bottom by the conjugate.  Example: Binomial denominator. Multiply top and bottom by

Dividing Radicals (part 2)  When rationalizing binomial denominators, multiply top and bottom by the conjugate.  Example: Binomial denominator. Multiply top and bottom

13 Question Quiz! Solve without a calculator Use your calculator to solve Simplify

7. Multiply. 8. Add or Subtract Divide. Solve

Answers!