Challenge. Exploring Patterns in the Units Digit of x n

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Holt McDougal Algebra 1

Challenge

Exploring Patterns in the Units Digit of x

n When you write out the first several powers of xn_{, where x and n are}

positive integers, you can discover interesting patterns in the units digits of xn_.

x1 _x2 _x3 _x4 _x5 _x6

x = 2 21₌₂₂2₌₂₍₂₎₌₄₂3₌₂₍₄₎₌₈₂4₌₂₍₈₎₌₁₆ ₂5₌₂₍₁₆₎₌₃₂₂6₌₂₍₃₂₎₌₆₄

Notice that 21_{and 2}5_{have the same units digit and that 2}2_{and 2}6_have

the same units digit. In the exercises that follow, you can discover other number patterns involving the units digits of xn_.

In Exercises 1–10, find the first nine powers of each value of x. Using the units digit of each result, complete the table. You may find a calculator useful. x1 _x2 _x3 _x4 _x5 _x6 _x7 _x8 _x9 1. x = 1 2. x = 2

2

4

8

6

2

4

3. x = 3 4. x = 4 5. x = 5 6. x = 6 7. x = 7 8. x = 8 9. x = 9 10. x = 10

Refer to the table that you completed in Exercises 1–10. Describe the pattern in the units digits of xn.

11. 1n_{______________________________________________________________________________________}

12. 2n_{______________________________________________________________________________________}

13. 3n_{______________________________________________________________________________________}

14. 5n_{______________________________________________________________________________________}

15. Write a rule that determines the units digit of 7n_{as a function of n.}

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7. 8 8. 125 9. 8 10. 8 11. 1 12. 1331 13. y 14. x2_y6 15. a2_b _16._x5 17. x2_y4 _18._x 19. 20 m Practice C 1. 9 2. 6 3. 10 4. 15 5. 8 6. 3 7. −3 8. 1024 9. 27 10. 9 11. 16 12. 1728 13. x2 _14._x8_y 15. y3_z4 _16._ab5 17. x 18. y 19. 54 cm2

Review for Mastery

1. 8 2. 10 3. 1 4. 4 5. 2 6. 7 7. 6 8. 14 9. 3 10. 1 11. 7 12. 5 13. 8 14. 8 15. 4 16. 1 17. 81 18. 1000 19. 4 20. 243 21. 8 22. 32 23. 343 24. 256 Challenge Problem Solving 1. 6 s 2. 51.3 mi/h 3. 512 in3 _{4. 4 cm} 5. D 6. G 7. A Reading Strategies 1. 3rd or cube 2. 5th 3. 5th; 4th 4. 2 5. 9 6. 2 7. 20 8. 4 9. 27 10. 8 11. 1024 12. 64 POLYNOMIALS Practice A 1. 2; 1 2. 3; 2 3. 5; 4 4. trinomial 5. monomial 6. binomial

Review for Mastery

1. 5; 2•2•2•2•2; ₃₂1 2. 10•10•10; ₁₀₀₀1 3. 4; 4; 5•5•5•5; 625 4. 5₄ y 5. 8a3 _6. 3 2 9x y 7. x4 y 8. ab 9. 5y₄2 x 10. 4 11. 512 12. 200 13. −8 14. 324 15. 1₁₆ Challenge 1. 1; 1; 1; 1; 1; 1; 1; 1; 1 2. 2; 4; 8; 6; 2; 4; 8; 6; 2 3. 3; 9; 7; 1; 3; 9; 7; 1; 3 4. 4; 6; 4; 6; 4; 6; 4; 6; 4 5. 5; 5; 5; 5; 5; 5; 5; 5; 5 6. 6; 6; 6; 6; 6; 6; 6; 6; 6 7. 7; 9; 3; 1; 7; 9; 3; 1; 7 8. 8; 4; 2; 6; 8; 4; 2; 6; 8 9. 9; 1; 9; 1; 9; 1; 9; 1; 9 10. 0; 0; 0; 0; 0; 0; 0; 0; 0

11. For all n, 1n_{has 1 as its units digit.}

12. The pattern is 2, 4, 8, and 6, for n = 1, 2, 3, and 4 and then repeats.

13. The pattern is 3, 9, 7, and 1, for n = 1, 2, 3, and 4 and then repeats.

14. For all n > 0, 5n_{has 5 as its units digit.} 15. If you divide n by 4, then the units digit is

7, 9, 3, or 1, depending on whether the remainder is 1, 2, 3, or 0, respectively. Problem Solving 1. 4₂₅ or 0.16 mm2 _{2. 3} 8 and 34 oz 3. 3.142 4. 42 2₃ liters 5. B 6. H 7. C Reading Strategies 1. 6 2. 0 3. 8−3 _4. 7 1 b 5. 32 6. ₃₂1 7. 1 8. _1,000,0001 9. −64 10. − 1 64 11. 1₄ t 12. 2 3 c d 13. 8₅ x 14. 12 RATIONAL EXPONENTS Practice A 1. B 2. D 3. C 4. A 5. 7 6. 3 7. 1 8. 12 9. 8 10. 9 11. 1 12. 32 13. x8 _14._x3_y4 15. m4_n _16._x2 17. 14 cm Practice B 1. 3 2. 11 3. 0 4. 11 5. 4 6. 8 A2

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Practice A

Integer Exponents

Simplify. The first one has been done for you. 1. 3−2= 1₂ 3 = ____ i ____ 1 3 3 = ___ 1 9 2. 2−4= 4 1 2 = ___ ___ ___ ___i i i 1 = ___1 3. (−3)−3= 3 ___ 1 ( ) = ___ ___ ___i i 1 = ______ 1 4. (−1)−5₌ 5 ___ 1 ( ) = ___ ___ ___ ___i i i i___ 1 = ___1 =___ 5. − 0 ____________________ (7.2) 6. −3 ____________________ (4)

Evaluate each expression for the given value(s) of the variable(s).

7. x−2_{for x}₌₃ _8._m0_n−3_{for m}₌_{2 and n}₌₃ _{9. 5r}−4_{for r}₌₋₂

(3)−2₌ 2 ___ 1 ( 3 ) = ___1

( )

0 ___

( )

___ −3 =

( )

___ • _{___}3 1 ( ) 5

( )

−4 ___ = 5 • _{___}− 1 ( ) = = 5 • i i i ___ ___ ___ ___ 1 = 5 • ____1 = Simplify. 10. 4x−3 _11. −2 5 b _________________________________________ ________________________________________ 12. k₂−4 13. f₋4₁ g _________________________________________ ________________________________________ 14. The weight of a silver charm is 2−2_grams.

Evaluate this expression. _____________________________________ evaluate:

find the value of

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Answer Key Exponents and Polynomials

INTEGER EXPONENTS Practice A 2. 2; 2; 2; 2; 16 3. −3; −3; −3; −3; −27 4. −1; −1; −1; −1; −1; −1; −1; −1 5. −1 6. 1₆₄ 7. 9 8. 2; 3; 1; 3; 1₂₇ 9. −2; (−2)4_{; −2; −2; −2; −2; 16; 5} 16 10. 4₃ x 11. 5b 2 12. 1₄ 2k 13. f 4_g 14. 1₄ gram or 0.25 gram Problem Solving 2. 3 8 and 34 oz 3. 42 2₃ liters 4. B 5. H 6. C RATIONAL EXPONENTS Practice A 2. D 3. C 4. A 6. 3 7. 1 8. 12 9. 8 10. 9 11. 1 12. 32 13. x8 _{14. 14 cm} Problem Solving 2. 9; 27; 51.3; 51.3 mi/h 3. C 4. G 5. A POLYNOMIALS Practice A 2. 3; 2 3. 5; 4 4. trinomial 5. monomial 6. binomial 7. 3x2₋_x_{+ 12; 3} 8. −g5₊_g4_{− 2}_g3_{; −1} 9. k4₋_k3₊_k2_{+ 1; 1}

10. quadratic monomial 11. linear binomial 12. quartic trinomial 13. −3; 2; 13; −2; 5 14. 72 in2 Problem Solving 2. 4; 360; −64; 360; 296 feet; 25; 900; −400; 900; 500 feet 3. h = 0.25: 0.9375 cubic feet; h = 0.5: 1 cubic foot 4. A 5. C

ADDING AND SUBTRACTING POLYNOMIALS Practice A 2. –20p5_{− 3}_p_{+ 14} _{3. 3}_m_{+ 6} 4. 5y2₊_y_{+ 12} _{5. 6}_z3_{+ 4}_z2_{+ 5} 6. 12g2_{+ 4}_g_{− 1} _{7. 8}_k_{+ 1} 8. 3s3_{+ 5}_s_{+ 20} _{9. 9}_a4_{+ 8}_a2 10. 9b2₊_b_{− 9} _11._w_{+ 8} 12. a. 2n + 2 b. 8n + 20 Problem Solving 2. 50x − 6; 94 yards 3. B 4. C MULTIPLYING POLYNOMIALS Practice A 2. 15x3 3. 18y5 _{4. 15}_x_{+ 21} 209

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Practice B

Integer Exponents

Simplify. 1. 5−3₌ ___ ______ 1 ₌ _____ 1 _{2. 2}−6₌ ___ ______ 1 ₌ _____ 1 3. 2 _______________________ ( 5)₋ − _4. 3 ______________________ (4)− − 5. 0 _________________________ 6 − 6. 2 ________________________ (7)−

Evaluate each expression for the given value(s) of the variable(s).

7. d−3_{for d}₌₋₂ _8._a5_b−6_{for a}₌_{3 and b}₌₂ _{9. (b}₋₄₎−2_{for b}₌₁

________________________ _________________________ ________________________ 10. 5z−x_{for z}₌₋_{3 and x}₌₂ _{11. (5z)}−x_{for z}₌₋_{3 and x}₌_{2 12. c}−3₍₁₆−2_{) for c}₌₄

________________________ _________________________ ________________________ Simplify. 13. t−4 _{14. 3r}−5 _15. 3 5 s t − − ________________________ _________________________ ________________________ 16. h₃0 17. 2x y3₄ 2 z − − 18. 4 ₃5 5 fg h − − ________________________ _________________________ ________________________ 19. 14 4₁ 20 a bc − − 20. 4 2 0 1 3 a c e b d− − 21. 2 2 0 3 6 g hk h − − − − ________________________ _________________________ ________________________ 22. A cooking website claims to contain 105_recipes.

Evaluate this expression. _____________________________________

23. A ball bearing has diameter 2−3_inches.

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Answer Key For Exponents and Polynomials

INTEGER EXPONENTS Practice A 1. 3; 3; 9 2. 2; 2; 2; 2; 16 3. −3; −3; −3; −3; −27 4. −1; −1; −1; −1; −1; −1; −1; −1 5. −1 6. 1₆₄ 7. 3; 9 8. 2; 3; 1; 3; 1₂₇ 9. −2; (−2)4_{; −2; −2; −2; −2; 16; 5} 16 10. 4₃ x 11. 5b 2 12. m₄3 n 13. 4 1 2k 14. f4_g _15._r6_s2 16. 1₄ gram or 0.25 gram 17. 10,000; 1000 Practice B 1. 5; 3; 125 2. 2; 6; 64 3. 1₂₅ 4. − 1 64 5. −1 6. 1₄₉ 7. −1 8 8. 24364 9. 1₉ 10. 5₉ 11. 1₂₂₅ 12. _16,3841 13. 1₄ t 14. _r35 15. t5₃ s 16. 13 17. _{3 2 4}2 x y z 18. 3 5 4 5 fh g 19. 7₄ 10 c a b 20. a 4_bc2_d3 21. _{2 2} 2 h g k 22. 100,000 23. 1₈ inch or 0.125 inch Practice C 1. 1₁₆ 2. 1 3. − 1 36 4. −1 5. 1₉ 6. 1₁₂₅ 7. − 1 343 8. −10241 9. 1 10. 27₁₆ 11. 5₂₇ 12. 1₈₁ 13. − 1 27 14. 18 15. 1₅₁₂ 16. 1₃ x 17. 1 18. 1₉ t 19. 3₂ n 20. 4 2 3x 21. − 1₂ a 22. 4 3 10s r 23. b c3 2₃ d 24. 2 3 5 x y 25. ₉s3_{4 2} p q r 26. 3 2 c b d 27. g kj3 ₂ 5 h 28. _{10,000 inch or 0.0001 inch}1 29. 1₂₁₆ inch or 0.00463 inch A1

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Practice C

Integer Exponents

Simplify. 1. 2 4 _________________− _2. 0 6 ___________________ 3. 2 6 ________________− − 4. 5 ( 1) _______________₋ − _5. 2 ( 3) ________________₋ − _6. 3 5 _________________− 7. 3 7 ________________− − 8. 5 ( 4) ________________₋ − _9. 0 ( 9) _______________−

Evaluate each expression for the given value(s) of the variable(s).

10. x−4_y3_{for x = 2 and y = 3} _{11. 5r}– 3_s−6_{for r = 3 and s = 1}

_________________________________________ ________________________________________ 12. (3 − m)−4_{for m = 6} _13._−2a−1_b−3_{for a = 2 and b = 3}

_________________________________________ ________________________________________ 14. (−2xy)−3_{for x = −2 and y = 1}

2 15. 3 4 5m − ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ for m = 10 _________________________________________ ________________________________________ Simplify. 16. x−3_{___________________ 17.}_z0_{____________________} _18._t−9_{____________________} 19. 3n−2_{__________________ 20.} 2 4 3x− _________________ 21. −a−2 ___________________ 22. 3 4 ________________ 10r s− _23. 3 2 3 _{_________________} b c d− 24. 2 3 0 _{_______________} 5x y z − − 25. p q₂9 ₃4 _{_________________} r s − − − 26. 0 2 3 _{_________________} a b c d − − 27. 3 2 1 5 _{_________________} g h k j − − −

28. A micrometer is an instrument that can measure the thickness of an object very accurately. One micrometer is accurate to within 10−4_inches.

29. An object is being measured by a micrometer. It has a

thickness of 6−3_{inches. Evaluate this expression.} _{_____________________________________}

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Answer Key For Exponents and Polynomials

INTEGER EXPONENTS Practice A 1. 3; 3; 9 2. 2; 2; 2; 2; 16 3. −3; −3; −3; −3; −27 4. −1; −1; −1; −1; −1; −1; −1; −1 5. −1 6. 1₆₄ 7. 3; 9 8. 2; 3; 1; 3; 1₂₇ 9. −2; (−2)4_{; −2; −2; −2; −2; 16; 5} 16 10. 4₃ x 11. 5b 2 12. m₄3 n 13. 4 1 2k 14. f4_g _15._r6_s2 16. 1₄ gram or 0.25 gram 17. 10,000; 1000 Practice B 1. 5; 3; 125 2. 2; 6; 64 3. 1₂₅ 4. − 1 64 5. −1 6. 1₄₉ 7. −1 8 8. 24364 9. 1₉ 10. 5₉ 11. 1₂₂₅ 12. _16,3841 13. 1₄ t 14. _r35 15. t5₃ s 16. 13 17. _{3 2 4}2 x y z 18. 3 5 4 5 fh g 19. 7₄ 10 c a b 20. a 4_bc2_d3 21. _{2 2} 2 h g k 22. 100,000 23. 1₈ inch or 0.125 inch Practice C 1. 1₁₆ 2. 1 3. − 1 36 4. −1 5. 1₉ 6. 1₁₂₅ 7. − 1 343 8. −10241 9. 1 10. 27₁₆ 11. 5₂₇ 12. 1₈₁ 13. − 1 27 14. 18 15. 1₅₁₂ 16. 1₃ x 17. 1 18. 1₉ t 19. 3₂ n 20. 4 2 3x 21. − 1₂ a 22. 4 3 10s r 23. b c3 2₃ d 24. 2 3 5 x y 25. ₉s3_{4 2} p q r 26. 3 2 c b d 27. g kj3 ₂ 5 h 28. _{10,000 inch or 0.0001 inch}1 29. 1₂₁₆ inch or 0.00463 inch A1

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Problem Solving

Integer Exponents

Write the correct answer.

1. At the 2005 World Exposition in Aichi, Japan, tiny mu-chips were embedded in the admissions tickets to prevent

counterfeiting. The mu-chip was developed by Hitachi in 2003. Its area is 42₍₁₀₎−2_{square millimeters. Simplify}

this expression.

_________________________________________ 3. Saira is using the formula for the area

of a circle to determine the value of π. She is using the expression Ar−2_where

A = 50.265 and r = 4. Use a calculator to evaluate Saira’s expression to find her approximation of the value of π to the nearest thousandth.

_________________________________________

2. Despite their name, Northern Yellow Bats are commonly found in warm, humid areas in the southeast United States. An adult has a wingspan of about 14 inches and weighs between 3(2)−3_{and 3(2)}−2_{ounces. Simplify these}

expressions.

_________________________________________ 4. The volume of a freshwater tank can

be expressed in terms of x, y, and z. Expressed in these terms, the volume of the tank is x3_y−2_{z liters. Determine}

the volume of the tank if x = 4, y = 3, and z = 6.

_________________________________________

Alison has an interest in entomology, the study of insects. Her collection of insects from around the world includes the four specimens shown in the table below. Select the best answer.

Insect

Mass

Emperor Scorpion 2−5_kg

African Goliath Beetle 11−1_kg

Giant Weta 2−4_kg

Madagascar Hissing Cockroach 5−3_kg

6. Many Giant Wetas are so heavy that they cannot jump. Which expression is another way to show the mass of the specimen in Alison’s collection? F −(2)4 kg H _{2 • 2 • 2 • 2}1 kg G 4 1 2 − ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ kg J 142 kg

5. Cockroaches have been found on every continent, including Antarctica. What is the mass of Alison’s Madagascar Hissing Cockroach expressed as a quotient? A −₁₂₅1 kg C 1₁₅ kg

B 1₁₂₅ kg D 125 kg

7. Scorpions are closely related to spiders and horseshoe crabs. What is the mass of Alison’s Emperor Scorpion expressed as a quotient? A −₃₂1 kg C 1₃₂ kg B 1₂₅ kg D 32 kg 6-9 LESSON

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Review for Mastery

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Reading Strategies

Using Patterns

Studying the patterns that are found in expressions with exponents can help you remember the rules for evaluating expressions with integer exponents. 4 3 2 1 0 –1 –2 –3 –4 3 = 3 • 3 • 3 • 3 = 81 3 = 3 • 3 • 3 = 27 3 = 3 • 3 = 9 3 = 3 3 =1 1 3 = 3 1 1 3 =_{3 • 3 9}= 1 1 3 =_{3 • 3 • 3 27}= 1 1 3 =_{3 • 3 • 3 • 3 81}= ⎫ ⎪ ⎪ ⎬ ⎪ ⎪⎭ ⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭

Note that the rules are the same when the base is a variable: b3₌_b_•_b_•_b _g0₌₁ _k−5₌

5

1

k _m1−3 = m3

Answer each question.

1. What is the base of the expression 6−4_? _{_________________}

2. What number can go in the box to make a true statement: 5 = 1? _________________ 3. Write the expression 1₃

8 with a negative exponent. _________________

4. What is the reciprocal of b7_? _{_________________}

Simplify each expression.

5. 5 2 ___________________________ 6. 5 2 ___________________________− 7. ₇0 ___________________________ 8. 10 __________________________−6 9. 3 ( 4) _________________________− 10. 3 ( 4) _________________________₋ − 11. 4 ___________________________ t− _12. 2 3 _________________________ c d− 13. ₈ 5 __________________________ x− _14. 0 12 __________________________r

Negative exponents: The answer is the reciprocal of the same expression with a positive exponent.

Zero exponent: The answer is always 1 (if the base is not 0; 00_{is undefined).} Positive exponents: The answer is the base multiplied by itself the number of times identified by the exponent.

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Review for Mastery

Integer Exponents

Remember that 23_{means 2 × 2 × 2 = 8. The base is 2, the exponent is positive 3.}

Exponents can also be 0 or negative.

Simplify 4−2. Simplify x2y−3z0.

4−2 _x2_y−3_z0

2

1

4 Write without negative exponents.

2 0 3

x z

y Write without negative

exponents. 1

4 • 4 Write in expanded form.

2 3 (1) x y z 0_{= 1.} 1 16 Simplify. 2 3 x y Simplify.

Fill in the blanks to simplify each expression.

1. 2−5 _{2. 10}−3 _3. 4 1 5− 2−5_{= 1} 2 10 −3_{= 1} 10 4 1 5− = 5 1₅ 2 = 1 3 1 10 = 1 _{5 =} =_{____________} =_{____________} =_{____________} Simplify. 4. ₅ 4 _____________ y− _5. 3 8 _____________ a− 6. 9x y3 −2 ____________ 7. x₁3 _{_____________} x y− 8. 2 1 3 _____________ b a b− 9. 5x y−4 2 _____________

Zero Exponents

Negative Exponents

in the Denominator

Definition

For any nonzero

number x, x0_{= 1.}

For any nonzero number x and any integer n, x−n_{= 1}

n x .

For any nonzero number x and any integer n,

1 n x− = xn.

Examples

₆0_{= 1} 1 0 2 ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ = 1 5 −3₌ 3 1 5 2 −4₌ 4 1 2 2 1 8− = 82 4 1 2− = 24

00_{and 0}−n_{are undefined.}

Review for Mastery

Integer Exponents

continued

Evaluate a−3b4 for a= 5 and b= 2.

a−3_b4

(5−3₎₍₂4₎ _Substitute. 4

3

2

5 Write without negative exponents. 16

125 Simplify.

When evaluating, it is important to determine whether the negative is raised to the power.

Evaluate −x−2_for_x₌_10. _{Evaluate (}₋_x₎−2_for_x₌_10.

−x−2 ₍₋_x)−2 −10−2 _Substitute. ₍₋₁₀₎−2 _Substitute. 2 1 10 − 1 ₂ ( 10)− 1 10 • 10

− Write in expanded form. _{( 10) • ( 10)}₋ 1 ₋ Write in expanded form. 1

100

− Simplify. ₁₀₀1 Simplify.

Evaluate each expression for the given value(s) of the variable(s).

10. x2_y0_{for x = −2 and y = 5} _11._a3_b3_{for a = 4 and b = 2}

_________________________________________ ________________________________________ 12. z3₂

y− for z = 2 and y = 5 13. −a3b−4 for a = 2 and b = −1

_________________________________________ ________________________________________ 14. n 2₄

m −

− for m = 6 and n = 2 15. (−u)2v−6 for u = 2 and v = 2

_________________________________________ ________________________________________ The negative is not raised

to the power. The negative is raised to the power.

Write without negative

exponents Write without negative exponents

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Name ________________________________________ Date ___________________ Class __________________

Review for Mastery

Integer Exponents

Remember that 23_{means 2 × 2 × 2 = 8. The base is 2, the exponent is positive 3.}

Exponents can also be 0 or negative.

Simplify 4−2. Simplify x2y−3z0.

4−2 _x2_y−3_z0

2

1

4 Write without negative exponents.

2 0 3

x z

y Write without negative

exponents. 1

4 • 4 Write in expanded form.

2 3 (1) x y z 0_{= 1.} 1 16 Simplify. 2 3 x y Simplify.

Fill in the blanks to simplify each expression.

1. 2−5 _{2. 10}−3 _3. 4 1 5− 2−5_{= 1} 2 10 −3_{= 1} 10 4 1 5− = 5 1₅ 2 = 1 3 1 10 = 1 _{5 =} =_{____________} =_{____________} =_{____________} Simplify. 4. ₅ 4 _____________ y− _5. 3 8 _____________ a− 6. 9x y3 −2 ____________ 7. x₁3 _{_____________} x y− 8. 2 1 3 _____________ b a b− 9. 5x y−4 2 _____________

Zero Exponents

Negative Exponents

in the Denominator

Definition

For any nonzero

number x, x0_{= 1.}

For any nonzero number x and any integer n, x−n_{= 1}

n x .

For any nonzero number x and any integer n,

1 n x− = xn.

Examples

₆0_{= 1} 1 0 2 ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ = 1 5 −3₌ 3 1 5 2 −4₌ 4 1 2 2 1 8− = 82 4 1 2− = 24

00_{and 0}−n_{are undefined.}

Name ________________________________________ Date ___________________ Class __________________

Review for Mastery

Integer Exponents

continued

Evaluate a−3b4 for a= 5 and b= 2.

a−3_b4

(5−3₎₍₂4₎ _Substitute. 4

3

2

5 Write without negative exponents. 16

125 Simplify.

When evaluating, it is important to determine whether the negative is raised to the power.

Evaluate −x−2_for_x₌_10. _{Evaluate (}₋_x₎−2_for_x₌_10.

−x−2 ₍₋_x)−2 −10−2 _Substitute. ₍₋₁₀₎−2 _Substitute. 2 1 10 − 1 ₂ ( 10)− 1 10 • 10

− Write in expanded form. _{( 10) • ( 10)}₋ 1 ₋ Write in expanded form. 1

100

− Simplify. ₁₀₀1 Simplify.

Evaluate each expression for the given value(s) of the variable(s).

10. x2_y0_{for x = −2 and y = 5} _11._a3_b3_{for a = 4 and b = 2}

_________________________________________ ________________________________________ 12. z3₂

y− for z = 2 and y = 5 13. −a3b−4 for a = 2 and b = −1

_________________________________________ ________________________________________ 14. n 2₄

m −

− for m = 6 and n = 2 15. (−u)2v−6 for u = 2 and v = 2

_________________________________________ ________________________________________ The negative is not raised

to the power. The negative is raised to the power.

Write without negative

exponents Write without negative exponents

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6-7 LESSON

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Review for Mastery

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Steps for Success

Step I The following ideas may help students better understand the examples in the lesson.

• Teach the lesson opener. Explain the pattern shown in the table. Have the students create their own table with another power. For example, a table showing the powers of 3 will provide a similar pattern.

• Remind students that some of the properties of exponents were taught in Chapter 1. Review these properties with the students.

Step II Ask English Language Learners to complete the worksheet for this

lesson.

• Problem 1 supports Example 2b and 2c in the student textbook. Make sure students understand the difference between a negative sign inside the parenthesis and a negative sign not in parenthesis. Give additional examples, such as 2(10 ) 2 and 21 0 2 , to emphasize how these two values differ.

• Problem 2 supports Example 4b in the student textbook. • Think and Discuss supports the worksheet.

Step III Teach the lesson.

• Explain to students that a negative exponent has nothing to do with whether the term is positive or negative.

Making Connections

• Use the Remember box next to the lesson opener in the student textbook to review the parts of a power with the students.

• Relate the negative sign in an exponent to a red flashing light that tells the students the power is in the wrong place. Once they move the power to the correct place, they can turn the red flashing light off by removing the negative sign.

• Group students and instruct them to create a worksheet of approximately 10 questions based on this lesson. Groups should create both the

worksheet and an answer key. The key should be turned in for you to

keep. Each group should exchange worksheets with another group and complete the worksheet for homework. The next day, the groups should then exchange the answer keys and check the homework.

• Write integers on index cards. Place a strip of magnet on the back of each card to use with a white board. Arrange the cards on the board as an expression with exponents. Allow students to come up to the white board and simplify the expression by rearranging the various cards.

Success for Every Learner

Integer Exponents

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Problem 1

Negative INSIDE the

parenthesis

(

2

3

)

24

Negative OUTSIDE

the parenthesis

2

3

24

5

2

(3

)

24

Problem 2

Think and Discuss

1. How do (23 ) 24 and 2 3 24 differ?

2. When you move a non-zero number with a negative exponent from the denominator to the numerator, what do you have to do to the exponent?

Success for Every Learner

Integer Exponents

(23 ) 24 5 (23 ) 21 ? (23 ) 21 ? (23 ) 21 ? (23 ) 21 5____ ₍ 1 23) ? ____ (21 3) ? ____ (21 3) ? ____ (21 3) 5 ____________________ ₍ 1 23) ? (23) ? (23) ? (23) 5___ ₈₁1 2 3 24 521 ? 3 24 521 ? 3 21 ? 3 21 ? 3 21 ? 3 21 521 ? 1 __ ₃? __ 1 ₃? 1 __ ₃? 1 __ ₃ 521 ? 1 ___ ₈₁ 52___ ₈₁1 A negative exponent in the denominator

moves the base

to the numerator. 24 ____ k 24 5 2___ 4 1 __ k 4 524 4 __ 1 k 4 524 ? k 4 __ ₁524 k 4

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Integer Exponents

1. The answer to (23 ) 24_{is positive and the}

answer to 2 3 24 is negative.

2. Make the exponent positive.

Rational Exponents 1. 3

2. The exponent in the exponential expression is the quotient of the exponent in the radical expression and the root index.

3. When m 5 1,



n b 1  5 b 1 __ n ₅n_{b .}

Polynomials

1. A monomial only has one term and a polynomial has many terms.

2. Find the degree of each monomial. The degree of the polynomial is equal to the degree of the monomial with the largest degree.

Adding and Subtracting Polynomials 1. Represent m 2_{with a square with side}

length equal to m.

2. You must first distribute the negative sign.

3. The Commutative Property of Addition

Multiplying Polynomials 1. The final product is x 2

2 3x 2 10.

2. There is 1 x 2_{-tile, 10 x-tiles, and 25 1-tiles.} 3. x 2 _x _x _x _x x 1 1 1 1 x 1 1 1 1 x 1 1 1 1 x 1 1 1 1

Special Products of Binomials

1. The middle term in a square of a sum is positive and in a square of a difference it is negative.

2. There are 3 terms.

3. There are 2 terms.

Answers for

Exponents and Polynomials

152

Holt McDougal Algebra x

x

CHAPTER X-6 X-5

152

Holt McDougal Algebra 1

6

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