laws of exponents and scientific...

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Laws of exponents and scientific notation Student Activity Sheet 1; use with Overview

Copyright 2014 Agile Mind, Inc. ® Content copyright 2014 Charles A. Dana Center, The University of Texas at Austin

of 1 With space for student work

1. REVIEW Samuel and Brianna are working on their homework together. They disagree about the answer to one of the problems. Samuel says that 103 = 30. Brianna says that 103 = 1000. Who is correct? Explain how you know.

2. REVIEW Rewrite each of the following as a single number, without any exponents.

a. 22 = b. 23 = c. 25 =

d. 32 = e. 33 = f. 35 =

g. 41 = h. 42 = i. 43 =

3. Is the equation true? Justify your answer.


Laws of exponents and scientific notation Student Activity Sheet 2; use with Exploring “Special exponents”


Page 1 of 5 With space for student work

1. Investigate the claim that 20 = 1 by examining patterns in powers of 2. a. Complete the table of values.

24

23

22

21

b. Analyze the patterns you see in the table. What patterns do you see in the exponents in the first column? What patterns do you see in the answers in the second column?

c. Based on the pattern you see in the table, what is the value of 20? Justify your answer.





2. Using the patterns you see in the table, extend it to find the value of 2-3.

24

23

22

21

20

2-3

3. How are and related?

4. What is equal to? Justify your answer in two different ways.





5. Write a general rule for negative exponents when the base is 2.

6. Investigate a different base to see if the pattern you discovered for a base of 2 still holds true. a. Complete the table to find values for powers of 5.

b. What do the patterns in the table indicate about the value of ?

c. What do the patterns in the table indicate about , if n is an integer?





7. REINFORCE Use repeated division to complete the tables for each base, b.

33 32 31 30 3-1 3-2 3-3

(-2)3 (-2)2 (-2)1 (-2)0 (-2)-1 (-2)-2 (-2)-3

(-3)3 (-3)2 (-3)1 (-3)0 (-3)-1 (-3)-2 (-3)-3

(-4)3 (-4)2 (-4)1 (-4)0 (-4)-1 (-4)-2 (-4)-3

8. Based on your work so far, how would you complete this statement?

For any number x, except zero, x = and x = .





9. REINFORCE Simplify the following.

a. b. c. d.

10. Evaluate each expression.

6-2

100

4-1

7-1

(-5)0


Laws of exponents and scientific notation Student Activity Sheet 3; use with Exploring “The multiplication law”



1. Is the equation true? Justify your answer in two different ways.

2. Use what you know about exponents to show that x2 • x3 = x5.





3. Complete these tables by evaluating each expression. Problem 22•28 23•27 24•26 25•25 26•24 27•23 28•22 210

Value

Problem 53•54 52•55 51•56 57

Value

Record your observations about the tables by completing the following statements.

coefficient base difference sum 4. Notice that the used in each of the problems in the top table is 2

and that the of the exponents is always 10.

5 sum 7 2 5. Notice that the base used in each of the problems in the bottom table is and

that the sum of the exponents is always .





6. Use what you know about exponents to express each of these products using a single base and exponent.

a.

b. 3-5 • 32

c. 4-2 • 4-1





7. Use what you know about exponents to simplify x •x3.

8. Use what you know about exponents to simplify y2 • y4.

9. Apply what you have learned about exponents to complete this table.

Problem Value 22 • 28

32 • 312 50 • 54 5-2 • 52 y3 • y4 y16 • y-4

10. Complete this statement of the Law of Exponents for Multiplication:

For any number x, except zero,

• = .

11. REINFORCE 12. REINFORCE 13. REINFORCE


Laws of exponents and scientific notation Student Activity Sheet 4; use with Exploring “Other exponent laws”



1. Solve this puzzle to evaluate (102)3.

10 102 103 105 106 102+3 102•3 103•2

2. Write an equivalent expression for (3-3)2 using a single exponent. Show why your answer is

correct.

3 33 3-3 3-1 36 3-6 32•-3 3-3•2





3. REINFORCE Rewrite 721 as a power of a power of 7.

4. Rewrite each expression in equivalent form, using a single exponent. a. 32 • 35 b. (32)4

5. Rewrite each expression in simpler form, using a single exponent. Show how you know

your answers are correct. a. 32 • 42 b. 23 • 53





6. In general, how would you rewrite xm • ym in simpler form using a single exponent? Explain why your answer is correct.

7. Complete the statements below to show how to rewrite a power of a power, and how to multiply unlike bases with like exponents. a. (xm)n = b. xm • ym =

8. Write each of these expressions as an equivalent expression using a single exponent. If an expression cannot be written using a single exponent, write “Not possible.”

Not possible 2-10 22 28 215 103 106 107 109 1015





9. REINFORCE These are some examples of student work with exponents. Consider each example and indicate whether it is correct or incorrect. For incorrect examples, explain and correct the errors.

a. (155)7 = 1512 b. 35 • 24 = 69 c. (4-2)2 =


Laws of exponents and scientific notation Student Activity Sheet 5; use with Exploring “The division law”



1. Use what you know about exponents to simplify .

2. Check the rule of using x = 3.

3. Simplify . Then, check your answer by substituting 4 for x in the original expression

and in your answer.





4. Complete these tables by evaluating each expression.

Problem 32

Value

Problem 23

Value

Record your observations about the tables by completing the following statements.

difference sum base coefficient 5. Notice that the used in each of the problems in the top table

is 3 and that the of the exponents is always 2.

exponents 2 coefficients -3 9 6. Notice that the base used in each of the problems in the bottom table is

and that the difference of their is

always 3.

7. Check the rule of using x = 2 and y = 3.





8. Simplify .

9. Can you apply the Law of Exponents for Division by completing this table?

Problem Value

10. Complete this statement of the Law of Exponents for Division:

For x not equal to 0, = .





11. REINFORCE

12. REINFORCE

13. REINFORCE

14. REINFORCE

15. REINFORCE


Laws of exponents and scientific notation Student Activity Sheet 6; use with Exploring “Scientific notation”



1. Complete the table to show each number written in factored form and in scientific notation.

Number in standard form Number in factored form Number in scientific notation

40,000 4 • 10,000 4 × 104

400

40 4 • 10 4 × 101

4

0.4

0.04

0.004

0.0004

2. Based on the patterns in the table, express 400,000 in scientific notation.





3. How can you write the number 20,000,000 using scientific notation? 4. How can you write the number 0.0003 using scientific notation? 5. How can you write the number 25,000 using scientific notation?

6. The moon is approximately 221,000 miles from the Earth. Write 221,000 in scientific

notation.

7. The average weight of the largest bone in the inner ear is approximately 0.00109 ounces.

Write 0.00109 in scientific notation.

8. The center of the Milky Way is 27,000 light years away from the Sun. A light year (9.5

trillion kilometers or 9,500,000,000,000 kilometers) is the distance a beam of light travels in one year. Express each of these numbers in scientific notation.





9. How would you find the distance in kilometers from our galactic center to the Sun?

10. What is the justification for each step outlined below?

Step Justification

2.7 x (104 x 9.5) x 1012

2.7. x (9.5 x 104) x 1012

(2.7 x 9.5) x (104 x 1012)

11. Perform the final computations to express the distance from the center of the Milky Way

to the Sun in kilometers. Be sure to express your final answer in scientific notation.





12. Express the quotient in scientific notation.

13. Stephanie is letting her hair grow longer. She notices that her hair grew 0.0099 meters during the 30 days in April. Can you use scientific notation in solving this puzzle to determine the approximate number of meters by which her hair grew each day in April?

-4 -3 -2 -1 1 3 3.3 9.9 29.7 × ÷

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