} 9. Simplify the expression. Write your answer using only positive exponents. the expression that represents the thickness of your friend s finger.

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1 Practice A Match the equivalent expressions p A. 36 B. 4 9 C (23) (29) (25) Simplify the expression. Write your answer using only positive exponents. 6. x m y a (3b) x 3 y x 24 y a 2 b x 23 y 25. Finger Thickness Your friend tells you that her finger is inch thick. Evaluate the expression that represents the thickness of your friend s finger. 26. Floor Tile The minimum recommended width of the space between 6-inch by 6-inch tiles is 2 22 inch and the maximum recommended width is 2 2 inch. Simplify the expressions for the minimum and maximum widths of the space between the 6-inch by 6-inch floor tiles. 27. Hole Punch Your hole punch makes holes in your paper that have a diameter of 4 2 inch. a. Write an expression for the area of one punched hole. Use the formula for the area of a circle A 5 πr 2. b. Your hole punch makes three holes in a page. Write an expression for the total area punched out of one sheet of paper. Algebra 27

2 Practice B (22) (26) p p Simplify the expression. Write your answer using only positive exponents. 3. x y (2b) (23m) a 2 b x 22 y (4x 24 y 2 ) (8mn 3 ) 0 2. c 23 d x 2 y x 26 4y x 23 y Paper A sheet of 67-pound paper has a thickness of 00 2 inch. a. Write and evaluate an expression for the total thickness of 5 sheets of 67-pound paper. b. Write and evaluate an expression for the total thickness of 2 3 sheets of 67-pound paper. 26. Frogs A frog egg currently has a radius of 5 2 centimeter. Write an expression using positive exponents for the volume of the frog egg. Use the formula for the volume of a sphere V πr Metric System The metric system has names for very small lengths. a. One micrometer is 0 3 times the length of one nanometer. One nanometer is 0 29 meter. Write one micrometer in meters. b. One femtometer is 0 3 times the length of one attometer. One attometer is 0 28 meter. Write one femtometer in meters. c. One centimeter is 0 0 times the length of one picometer. One picometer is 0 22 meter. Write one centimeter in meters. 28 Algebra

3 Practice C p p (5 2 ) p Simplify the expression. Write your answer using only positive exponents. 0. (4x 23 y 4 ) 22. 9x 24 y x 4 y (4x 25 ) (22d 2 ) (2x) 24 y 8 2x 5 y x 26 y 4 (23x2 ) 24 y x 3 y 24 (4x 24 (2x 24 y 2 ) 2 8. y 7 ) 2 24x 26 y 2 Tell whether the statement is true or false for all nonzero values of a and b. If it is false, give a counterexample. 9. a 25 a 26 5 a 20. b 2 a 2 5 a b 2. a 2 b 2 5 a b 22. Guitar The world s smallest guitar is only 0 26 meter tall. An average guitar is about 0 0 meter tall. How many times taller is an average guitar than the world s smallest guitar? 23. Knitting Needles A size knitting needle has a diameter of about 4 2 centimeter and a size 8 knitting needle has a diameter of about 2 2 centimeter. a. How many times larger is the diameter of a size 8 needle than the diameter of a size needle? b. Suppose that each needle is 4 inches long. Write expressions for the approximate volume of each size of knitting needle. Use the formula for the volume of a cylinder V 5 πr 2 h. c. How many times larger is the approximate volume of a size 8 needle than the approximate volume of a size needle? d. Are your approximations in part (b) overestimates or underestimates? Explain your reasoning. Algebra 29

4 Challenge Practice In Exercises 5, a and b are real numbers such that a > 0 and b > 0. Tell whether the statement is always true, sometimes true, or never true. If it is sometimes true, give a set of values for which it is true and a set of values for which it is false.. a 23 b 24 5 b4 a 3 2. (a b) 22 5 a 22 b (a 2 b 2 ) /2 5 a b 4. (a 2 b 2 ) 2 5 a 4 2a 2 b 2 b 4 5. a b 5 a 2 b 2 6. Determine which positive values of a make a 23 > a 24 a true statement. In Exercises 7 0, evaluate the given expression for the given values of a. 7. [(a ) 22 ] 3 [(a 2 ) 23 ] 2; a [(a 2 3) a 2 2 ] 3 [(a 2 )2 ] 4 ; a [(a 2) a ] a a ; a (a a ) 22a (a ) 2a ; a 5 2 Algebra 35

5 Study Guide GOAL Use zero and negative exponents. EXAMPLE Use definition of zero and negative exponents a Definition of negative exponents 5 64 Evaluate exponent. b Definition of zero exponent c Definition of negative exponents Evaluate exponents Simplify. Exercises for Example EXAMPLE (25) Evaluate exponential expressions Evalute the expression. a. 3 6 p Product of powers property Subtract exponents Evaluate power. b. [(22) 24 ] 2 5 (22) 24 p 2 Power of a power property 5 (22) 28 Multiply exponents. 5 (22) Evaluate power. Definition of negative exponents 30 Algebra

6 Study Guide continued Exercises for Example (24) 7 p (24) EXAMPLE 3 Use properties of exponents Simplify the expression. Write your answer using only positive exponents. a. (3m 22 n 3 ) p (m 22 ) 3 p (n 3 ) 3 Power of a product property 5 27 p m 26 p n 9 Power of a power property 5 27n9 m 6 Definition of negative exponents b. (25st) 2 t 24 20s 3 t 5 (25st)2 t 8 28 Definition of negative exponents 20s 3 4 t 5 (25s2 t 2 )t 8 20s 3 t 4 Power of a product property 5 25s2 t 0 20s 3 t 4 Product of powers property 5 5t6 22s Quotient of powers property Exercises for Example 3 Simplify the expression. Write your answer using only positive exponents. 9. (5x 2 y 23 z) m 22 np 3 2m 2 n 25 p (2r 2 t) 23 rst 4 6r 6 s 23 Algebra 3

7 S 8. Problem Solving Workshop: Mixed Problem Solving For use with pages Multi-Step Problem A department store sells plastic cubical containers that can be used to store food. 4. Open-Ended The table shows units of measurement of length and their equivalents in meters. a. One of the containers has a side length of inches. Find the container s volume by writing the side length as an improper fraction and substituting the length into the formula for the volume of a cube. b. Identify the property of exponents you used to find the volume in part (a). Name of unit Length (meters) Terameter 0 2 Kilometer 0 3 Centimeter 0 22 Micrometer Multi-Step Problem There are about 0 3 white corpuscles in cubic millimeter of blood. a. Copy and complete the table by finding the number of white corpuscles for the given amounts of blood (in cubic millimeters). a. Use the table to write a conversion problem that can be solved by applying a property of exponents involving products. b. Use the table to write a conversion problem that can be solved by applying a property of exponents involving quotients. Blood (cubic millimeters) Number of white corpuscles 0? 00? 000? 0,000? 00,000? b. A particular sample of blood is 95,000 cubic millimeters. Use order of magnitude to write an expression you can use to find the approximate number of white corpuscles in the sample of blood. Simplify the expression. Verify your answer using the table. 3. Short Response A carrot seed has a mass of about 0 24 gram and is 0 3 times less massive than a sweet corn seed. A student says that a sweet corn seed has a mass of about gram. Is the student correct? Explain. 5. Gridded Answer The mass of a grain of sand is about 0 23 gram. About how many grains of sand are in a bag of sand that weighs 2.8 grams? 6. Extended Response For an experiment, a scientist dropped about 0 24 cubic inch of olive oil into a container of water to see how the oil would spread out over the surface of the water. The scientist found that the oil spread until it covered an area of about 0 22 square inch. a. About how thick was the layer of oil that spread out across the water? Check your answer using unit analysis. b. The water has a surface area of 0 2 square inches. If the oil spreads to the same thickness as in part (a), how many cubic inches of olive oil would be needed to cover the entire surface of the water? c. Explain how you could find the amount of oil needed to cover a container of water with a surface area of 0 x square inches. Algebra 33