You discovered in Lesson 4.1 that when two powers with the same base are multiplied, the base remains the

Transcription

1 Division Properties of Exponents Lesson 4.2 You discovered in Lesson 4.1 that when two powers with the same base are multiplied, the base remains the same and the exponents are added together. Examine the division problems below. 5⁷ 5³ = = 5⁴ x⁶ x⁴ = x x x x x x x x x x = x² What is the relationship between the original exponents and the resulting exponent? 5⁷ 5³ = 5⁴ x⁶ x⁴ = x² When two powers with the same base are divided, the base remains the same and the exponents are subtracted. You can also simplify a power of a quotient or fraction by distributing the power to both the numerator and denominator. ( 3_ 4 ) ³ = 3_ 4 3_ 4 3_ 4 = 3³ 4³ = Lesson 4.2 ~ Division Properties of Exponents 129

2 Example 1 Solutions Simplify each of the following. a. x⁵y⁸ x²y b. ( 3m⁴ 5 ) ² a. Group powers that have the same base. ( x ⁵ Subtract exponents. x ² ) ( y ⁸ y ) x ⁵ ² y ⁸ ¹ = x ³ y ⁷ b. Distribute the power to each base. ( _ 3m ⁴ 5 ) ² = _ 3 ² ( m ⁴ ) ² 5 ² Multiply powers of powers and _ 9m ⁸ 25 evaluate coefficients. EXPLORE! going Negative Step 1: Copy the tables below. Find the value of each power with a calculator. If the value of the power is less than 1, write the power as a fraction. Power 2⁴ 2³ 2² 2¹ Value Power 2 ⁴ 2 ³ 2 ² 2 ¹ Value Step 2: What do you notice about the powers with opposite exponents (i.e. 2² and 2 ²)? Step 3: Use your observation from Step 2 to predict the value of each power below. a. Given that 4² = 16, what is the value of 4 ²? b. Given that 3⁵ = 243, what is the value of 3 ⁵? c. Given that 6 ³ = 1, what is the value of 6³? 216 Step 4: Look at the statements below. What is the value of each expression (written without an exponent)? a. 5² 5² = =? b. 2³ Step 5: Notice that 5² 5² = 5² ² = 5⁰, 2³ 2³ 2³ =? c. 3⁴ 3⁴ =? = 2⁰ and 3⁴ = 3⁰ using the Division Property of Exponents. 3⁴ Based on your findings in Step 4, what is the value of 5⁰, 2⁰ and 3⁰? Step 6: Use your calculator to raise other numbers to a power of 0. Try whole numbers and decimal values. What can you conclude? 130 Lesson 4.2 ~ Division Properties of Exponents

3 An expression is in simplest form only if it has no negative or zero exponents. As shown in the blue box above, an exponent with its base can be moved to the opposite side of the fraction bar to change its sign. Example 2 Solutions Simplify each of the following. a. _ ( 4p⁵k 3 ) ⁰ b. _ x²y- ⁴ z - ³ a. Any term to the power of 0 equals 1. _ 4p ⁵k ( 3 ) ⁰ = 1 c. 6a²b - ⁶c⁸ 2a²b - ⁵c - ¹ x ² y b. Write as separate factors. ( ⁴ z ³ ) = ( Use the rules for negative exponents. ( x ² 1 ) ( 1 Multiply factors. x ² z ³ _ y ⁴ y ⁴ ) ( x ² 1 )( z ³ 1 ) y ⁴ 1 )( 1 z ³ ) c. Write as separate factors. Group like bases. 6a ² b ⁶c ⁸ 2a ² b ⁵ c ¹ = ( 6 2 ) ( a ² b ⁶ a ² ) ( b ⁵ ) ( c ⁸ c ¹ ) Subtract exponents on like bases. 6 ( 2 ) ( a² ² )( b - ⁶ -(- ⁵ ) )( c⁸ -(- ¹ ) ) ( 3 )( a ⁰ )( b ¹ )( c ⁹ ) Use rules for zero and negative ( 3 ) ( 1 ) ( 1_ b ) c⁹ exponents. Multiply. 3c ⁹ _ b Lesson 4.2 ~ Division Properties of Exponents 131

4 Exercises For Exercises 1 4, write TRUE or FALSE to indicate if the expression has been simplified correctly. 2x² 1. 6x ³ = x⁵ x³ 4 2. ( 3 ) ² = x⁵ 5x⁹ 9 3. x⁴ ( = 5x⁵ 4. p ¹ ) ² = 25p⁶ 5p² 5. 8¹² 8⁵ 6. x⁵ x² 7. a⁶b⁹ a³b⁵ 8. 2w⁵v⁴ 10wv² 9. ( d ² g ³ ) ⁵ 10. ( 2y ³ 3 ) ³ 11. ( 4yh² )⁰ 12. ( 3w ⁴ ) ⁰ ² 5x¹¹ ⁴ 15. k ³m² n ⁷ 16. 7p ²q ⁵ 17. A contractor plans to build a rectangular office complex. The area of the first floor of the complex is represented by 15x ⁵y ⁹. The length of the complex is represented by 3x ²y ⁵. What expression represents the width? Show all work necessary to justify your answer. 18. Write and solve a problem that requires you to subtract exponents. 19. A truckload of cement weighs approximately 4⁷ pounds. The driver of the truck weighs 4⁴ pounds. The truckload of cement weighs how many times more than the driver? Explain how you know your answer is correct. 20. Ebizah needed to find 8³ 8. His work is at the right. Ebizah made a mistake. Identify his mistake and find the correct quotient. 21. Alan's teacher asked every student to find three different expressions that simplify to 6x²y. Alan wrote 12x²y². Write two other expressions that simplify to 6x²y. 2y Ebizah's Work 8³ 8 = 8³ 8 = 1³ = ( 3m ⁵ m ¹¹ _ r ² t ⁰ 23. n ⁵ ) ³ 26. ( 24x ⁷y ⁴ 4x ³y ² 6y ² 24. z ³ ) ² 27. ( 10p ²w ⁶ 6p ²w ⁶ 2q ² w ³ 3x ⁴ y ⁵ ) ¹ 132 Lesson 4.2 ~ Division Properties of Exponents

5 Find the missing dimension when given the volume. Write the missing dimension in simplest form. 28. Volume = 10x ⁴ y ⁵ z 29. Volume = 14a ³ m ⁸ p ¹ 2x ²? y ²? 7p 2a ⁸m ³ review 30. p³p⁹ 31. (x⁵)³ 32. (2cd)³ 33. (2x²)² (3x²)³ 34. (5y³)² (y⁴)³ 35. ( 1_ 2 m) ²(2m²)³ 36. The length of a rectangle is three times its width. If its width is 2x² units, what is the term for the area of the rectangle? 37. The width of a rectangular prism is twice its length. The height is three times its length. If its length is p units, what is the term for the volume of the prism? 38. Nicholas graphed a triangle with vertices of (2, 4), (2, 0) and (5, 0). His classmate, Suzi, graphed a similar triangle with a perimeter of 36 units. What was the area of Suzi's triangle? Show all work to support your answer. Tic-Tac-Toe ~ M atch i ng Ga m e Write 15 non-simplified variable expressions that contain exponents. Make sure you have a minimum of five expressions containing fraction bars. Cut thicker paper (such as cardstock, construction paper, index cards or poster board) into 30 equal-sized pieces. Write each expression on one card. Write the expression in simplest form on another card. Use these cards to play a memory game with a friend, classmate or family member. Record each pair of cards that each participant wins on a sheet of paper by listing the non-simplified and simplified expressions. Give the cards and the game sheet to your teacher. Lesson 4.2 ~ Division Properties of Exponents 133