Algebra 1 Unit 6 Notes

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1 Algebra 1 Unit 6 Notes Name: Day Date Assignment (Due the next class meeting) Monday Tuesday Wednesday Thursday Friday Tuesday Wednesday Thursday Friday Monday Tuesday Wednesday Thursday Friday Monday Tuesday Wednesday Thursday 01/1/14 (A) 01/14/14 (B) 01/15/14 (A) 01/16/14 (B) 01/17/14 (A) 01/21/14 (B)) 01/22/14 (A) 01/2/14 (B) 01/24/14 (A) 01/27/14 (B) 01/28/14 (A) 01/29/14 (B) 01/0/14 (A) 01/1/14 (B) 02/0/14 (A) 02/04/14 (B) 02/05/14 (A) 02/06/14 (B) 6.1 Multiplying Exponents 6.2 Dividing Exponents 6. Negative Exponents 6.4 Simplifying Radicals 6.5 Product Property of Radicals 6.6 Simplifying Radicals with Multiple Terms 6.7 Cube Roots Practice Test Unit 6 Test NOTE: You should be prepared for daily quizzes. Every student is expected to do every assignment for the entire unit, or else Homework Club will be assigned! Student who complete 100% of the assignments for the semester will receive a 2% bonus! HW reminders: If you cannot solve a problem, get help before the assignment is due. Need help? Try or

2 6.1 Multiplying Exponents Essential Question: How do we simplify expressions by applying exponent properties involving products? Warm-up: 1. Graph y = 2. Graph y = 1. Graph y = 4 With your teacher, graph each line over the specified domain: 1. If x < 2 2. If x 0. If -1 x < 7 Vocabulary: Coefficient: 4x x Base: 4x x Exponent: 4x x What does x 4 mean? What does x mean? Simplify the following expression, x 4 x.

3 Product-of-Powers Property: For all nonzero numbers x and all integers m and n, x m x n = x ( + ) ***Note: This only works when the bases are the same. When multiplying two terms with the same, we keep the base and the exponents. Examples: Simplify each of the following expressions. Leave in exponential form x x 6. y 4 y 4. z 7 z 9 5. (x 2 y )(x y 6 ) 6. 2y 2 z 4 y 9 z 6 Objective #1: Can you find the product of terms with the same base? a. x x 8 x b. 4a 5 6a 7 c. y z 2 xy 5 z 2 Reflect: What are 2 different ways to write out the expression x yx 5 y 7 in order to simplify it?

4 What does (x 4 ) mean? Simplify (r 5 ) 2. Power-of-a-Power Property: For all nonzero number x and all integers m and n, (x m ) n = x ( ) Examples: Simplify each of the following expressions. Leave in exponential form. 1. (x 2 ) 2. (y 5 ) 4. (2 ) 2 Objective #2: Can you simplify an expression to more than one exponent? a. ( 4 ) 2 b. (x 5 ) 2 c. (x) What does (rs) mean? Power-of-a-Product Property: For all nonzero numbers x and y, and any integer n, (xy) n = x y Examples: Simplify each of the following expressions. Leave in exponential form. 1. (xy) 2. (b) 5. ( 4x) 2

5 Objective #: Can you simplify a product to an exponent? a. (x) 4 b. (x 2 z ) 5 c. ( 5a 4 ) 2 Example: Now try using all three properties in the same problem. 1. (xy 2 ) x 2 2. ( 5vw 4 ) 2 2vw Objective #4: Can you use all three properties in the same problem? a. (4a b 2 ) 2 (a) b. ( 4r 5 s) 2 2rs Objective #5: Can you explain and fix the mistakes made in the following problems? a. (y ) 8 = y 11 b. 2w 5 w 2 ( w 2 ) 4 = 6w 7 9w 8 = 54w 56

6 6.2 Dividing Exponents Essential Question: How do we simplify expressions by applying exponent properties involving quotients? Warm Up: 1. Graph y = x Graph y = 1 x 2. Graph y = 2x 2 With your teacher, graph each line over the specified domain: 1. If x < 2 2. If x 0. If -1 x < 5 What does mean? Quotient-of-Powers: For all nonzero numbers a and any positive integers m and n, m > n, a m an = a( ) ***Note: This only works when the bases are the same. When dividing two terms with the same, we keep the base and the exponents.

7 Examples: Simplify each of the following expressions = 6 = = = 4 = 4. ( 2)7 ( 2) y 9 y12 5. x 6 x x 2 Objective #6: Can you find the quotient of two terms with the same base? a. h5 h 2 ( 5)2 b. c. m m 5 ( 5) 17 m 7 Reflect: Can you explain the difference between simplifying a product with the same base and simplifying a quotient with the same base?

8 What does ( 4 )2 mean? Power-of-a-Quotient Property: For all real numbers a and b, b 0, and a positive integer m, ( a b )m = a b, b 0 Examples: Simplify each expression. 1. ( 4 7 ) = (r s )5 =. ( 4 w ) 4. ( 5 4 ) 2 5. ( 5 t )4 6. ( 2y7 xy 5) 7. ( x2 w )2 2 w Objective #7: Can you simplify a quotient to an exponent? a. ( 5 9 )2 b. ( x y )5 c. ( r r 2 )4 d. ( x2 y y 2 ) e. 1 4x (2x5 y )

6. Zero and Negative Exponents Essential Question: How do we simplify expressions by defining and using zero and negative exponents? Warm Up: 1. Graph y = x if x > 2 2.

9 6. Zero and Negative Exponents Essential Question: How do we simplify expressions by defining and using zero and negative exponents? Warm Up: 1. Graph y = x if x > 2 2. Graph y = 1 x if 6 x x x 2 x 1 = x x 0 x 1 x 2 1 x x x Definition of Zero and Negative Exponents a to the power a 0 =, a = a n is the reciprocal of a n a n = 1 an, a a n is the reciprocal of a n 1 a n = an, a 0 = = ***Remember, when dividing two terms with the same base, we subtract the exponents.

10 Examples: a6 = a a a a a a a 4 a a a a = a a 1 = or a 6 a 4 = a6 4 = a 6 = = = or a 6 a 6 a 6 = a = a = a 4 = = = or a 4 a 6 Examples: Simplify each quotient. Be sure all exponents are positive. a 6 = a = a = 1. x x 7 x 2. x. x 6 4. ( 2) x 9 6. x5 x 5 7. x 2 x 4 y 2 8. x 5 Special Cases: 9. ( 1 4 )

11 Example: Simplify the following expressions. Write your answer using only positive exponents. 1. 2w x (2wx) 2 6fg f 2 g. (yz 2 ) 2 Reflect: Summarize the order in which you should simplify when there are exponents in a problem? Objective #8: Can you simplify expressions involving zero and negative exponents? a. 0 b. x 6 4x 6 c. ( 5n6 n 2 n 4) d. 4 e. x x 4 f. (2 5 )

12 6.4: Simplifying Radical Expressions Essential Question: Can you simplify expressions that involve radicals? Warm Up: 1. Simplify (2x 2 y 5 ) 2 2. Graph y = 2 x if 6 x < 9 Squaring a number means multiplying a number by. The numbers 1, 4, 9, 16 are examples of, because 1 = 1 2, 4 = 2 2, 9 = 2, 16 = 4 2, etc. Make a list of perfect squares below (up to 225): The inverse of squaring is finding a.

13 Example 1: Simplify each expression. a) 49 b) 64 c) A square television set has an area of 144 square inches. Find the length of one side. Estimating Radical Expressions: What does the expression 6 mean? We are trying to find a number that we can to equal 6. Sometimes the numbers inside a radical,, will not be perfect squares. When this happens we need to the value of the expression. Examples: Estimate the value of each radical expression A square piece of paper has an area of 80 square inches. Which of the following is the best estimate for the length of one side of the paper? A) 9.1 inches C) 40 inches B) 8.9 inches D) 9 inches

14 Objective #9: Can you estimate the value of a radical expression? a. 60 b. 8 c. 201 **d. Which integer is closest to the value of 5? A. B. 8 C. 11 D. 15 **e. The value of 180 is between what two integers? A. 1 and 14 B. 9 and 42 C. 60 and 61 D. 2 and 24 Simplest Form of a Radical Expression: A radical expression is in simplest form if: a) no are factors of the inside. b) no are in the of a fraction. Product Property of Radicals: The square root of a product equals the of the of the factors. ab = a b Example: 20 = 4 5 = 2 5 ***Note that there are no factors of 5 that are perfect squares.

15 Examples: Simplify each of the following radical expressions Objective #10: Can you simplify radical expressions? a. 24 b. 90 c. 72 **d. 2 2 **e f. Describe the error in the problem below. Then work the problem correctly. 45 = 15 = 5 = 5

16 Radical Expressions involving Variables: What does x 2 mean? What does x mean? Example: Simplify 12x 2 Examples: Simplify each of the radical expressions x 2. 12x 2 y z 4 Objective #11: Can you simplify radical expressions involving variables? a. x 8 b. h c. x x 2 **d. x x **e. 50x y 5 Reflect: Find x 7 without writing out 7 x s.

17 6.5: Product Property of Radicals Essential Question: Can you simplify the product and quotients of radical expressions? Warm up: 1) Simplify: x x 5 2) Simplify: c 6 c 2 x + y = 9 ) Given the system of equations { y = 2x + 4 system? What is the solution for y in the Product Property of Radicals: The product of two radicals equals the of the. a b = ab Example: 2 8 = 16 = 4 Examples: Find the product and simplify each radical expression ( 21) 5. x 5 x

18 Objective #12: Can you simplify products of radical expressions? a. 6 8 b. 5(6 7) c. 2 x x 5 d. x 2 8 **e. ( 25x)( 4x ) f. Describe the error in the problem below. Then work the problem correctly = 20 = 4 5 = = 2 5 Quotient Property of Radicals: The square root of a quotient equals the of the of the numerator and denominator. a b = a b where a 0 and b > Reduce the fraction 2. Square root the top and bottom

19 Examples: Simplify each of the radical expressions x 18x 2 Objective #1: Can you simplify quotients of radical expressions? a b. 8 9 c d e **f. 12x 4x 2 ****Note: All of the examples above simplified to a whole number on the denominator. Sometimes that will not be the case. When the denominator is not a whole number, we have to rationalize the denominator.

20 Rationalizing the Denominator: The process of a from an expression s denominator = 2 11 rationalize! We cannot leave the radical in the, so we must Examples: Simplify each radical expression and be sure to rationalize the denominator Reduce the fraction 2. Simplify any radicals. Rationalize the Denominator. 16a a x

21 Objective #14: Can you simplify quotients of radical expressions and rationalize the denominator? a. 2 b c d. 24 e. 20b 14b **f b Reflect: How do you know when to rationalize the denominator of a radical expression? 6.6 Simplifying Radical Expressions with more than one term Essential Question: Can you simplify radical expressions using the distributive property and by combining like terms? Warm up: Simplify 1) x 2 + 5x x 2) 4y 2x + 6y ) 252 4) 24

22 Combining Like Terms: If two or more expressions have the same radicand, the stuff inside the square root, then we can combine like terms just like variables. Example: We can also simplify radicals to possibly create like terms. Example: Simplify each term Simplify each of the following radical expressions. 2. Combine like terms

23 Objective #15: Can you combine like terms when radical expressions are involved? a b c d Reflect: Define like terms for radical expressions. Distributing Radical Terms: We can use the property with radicals! Example: 2( 4 5) Simplify each of the following expressions. 1. (4 2 ) 2. 7( )

24 . 2x( + x) 4. 5x( x- 5) Objective #16: Can you distribute with radical expressions? a. 5( + 12) b. 5(2 2 5) Cube roots and simplifying radicals Essential Question: Can you estimate cube roots and simplify problems involving radicals? Warm up: Estimate the value of the following radicals: 1) 0 2) 70 2) Simplify [(5 2) 2 + ]

25 Cubing a number means multiplying it by itself times. The inverse of cubing a number is finding a. Cubes Equal Factors Cube Root 2 = = 4 = 5 = 6 = = Example: Estimate which two whole numbers a cube root is between. = 2 1) 20 2) 80 ) 100 Example: Order the expressions from least to greatest. 1) Order the expressions from least to greatest: 49, 64, 75

26 a) The value of 55 Objective #17: Can you estimate cube roots? is between which two integers? A. 2 and B. and 4 C. 4 and 5 D. 6 and 7 a) Which answer shows the expressions ordered from least to greatest? 64, 2, 27, 100 A. 64 C. 2, 27, 2, 27, 100, 64, 100 B. 2, 64 D. 100, 100, 27, 64, 27, 2 Simplifying Radicals: Make sure to use the correct order of operations! 1) Parenthesis 2) Exponents ) Add/Subtract 4) Square Root Example: 1) ( 1) 2 + (2 + 1) 2 2) (5 7) 2 + (1 + 1) 2 Objective #18: Can you simplify expressions with exponents and radicals? a) (6 5) 2 + ( + 1) 2 b) (2 8) 2 + (5 + ) 2

Chapter 2. Real Numbers and Monomials. 8/2016 LSowatsky 1

Chapter 2. Real Numbers and Monomials. 8/2016 LSowatsky 1 Chapter 2 Real Numbers and Monomials 8/2016 LSowatsky 1 2.1.A Powers and Exponents Main Idea: Use powers and exponents to write large and small numbers. LSowatsky 2 LSowatsky 3 Example: Write each expression