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2-6-1 Introduction to Exponents Repeated Multiplication Remember multiplication is a way to write repeated addition. To say 3+3+3+3 we write 3x4. Sometimes multiplication is done over and over and over. To write 3x3x3x3 we write 34. 42 (read “four squared”) means 4x4 so 42 =16 53 (read “five cubed”) is 5x5x5=125 Practice: Write the indicated operation using exponents. Then perform the operation. a) 4x4x4= 10X10X10X10X10X10= 2x2x2x2x2x2x2x2= b) c) d) e) f) 5x5= 0.3 x 0.3 x 0.3= 0.2 x 0.2 x 0.2 x 0.2= -3 x (-3) x (-3) x (-3)= aaaaa= (answer: a5) 3x3x3x3x3= 5.1 x 5.1= 0.01x0.01x0.01x0.01= -100x(-100)x(-100)= xxxxxxxxxxx= 9x9x9= (-1.2)(-1.2)(-1.2)= 1.5x1.5= (-1)x(-1)x(-1)x(-1)x(-1)= 2f2f2f2f= Practice: Perform the indicated operation. g) 25= h) i) j) k) 122= (-4)2= (-4)3= (-10)3= 72= 152= (-10)2= (-12)2= (0.5)4= 82= 252= (-13)2= (-5)3= (0.2)4= 1 Review Topic 7: Fraction Multiplication and Division Look at the following examples. Decide what the rule is for multiplying fractions. Examples: 2/3 x 1/5 = 2/15, 3 3 9 5 5 3 15 7 3 1 , 4/5 x 3/ 4 =12/20=3/5 , 8 1 8 8 4 4 16 8 Notice: Fraction numbers must be written as the ratio of two integers to be multiplied. Rule: a c ac b d bd Thus to multiply 2 ½ x 1 ¾ x 12 1. First change the mixed numbers to improper fractions 5/2 x 7/4 x12/1 2. to get 420/8 3. This can be simplified to 52 ½. One other step can save time. Canceling reduces the answer before you multiply the numbers. 15 7 12 15 7 12 3 7 7 147 14 7 10 2 4 25 2 4 25 5 2 1 5 10 1 Fraction Division To divide fractions, invert the second fraction and multiply. 2/3 ½ = 2/3 x 2/1 = 4/3 = 1 1/3. Notice: DON”T cancel until after the divisor (second number) is inverted. For word problems, be very careful which number to write first in the problem. The thing being split, cut apart, or sorted goes first. 3 3 Rule: a c a d ad x b d b c bc 1 1 10 5 10 2 2 2 4 1 3 2 1 3 2 3 2 3 5 3 1 3 3 1. 2. 3. 4. 5. Change to improper Invert and Multiply Cancel Multiply Simplify Practice: a) 1 2 = 5 3 1 2 = 5 3 1 2 = 5 3 1 2 = 5 3 1 2 = 5 3 1 2 = 5 3 1 2 = 5 3 b) 1 2 = 5 3 c) 2 3 2 = 2 3 2 = 2 3 3 = 2 3 4 = 2 d) 1 3 3 5 1 = 4 7 5 3 3 4 3 8 4 53= 1 3 3 5 1 = 4 7 5 3 3 4 3 8 4 (-3)4= 1 3 3 5 1 = 4 7 5 3 3 4 3 8 4 (-8)2 e) 3 3 4 3 8 4 (-2)7= f) g) h) Jack has 2 ¾ sacks of flour. If he uses 2/3 of the flour, how much flour is left? 15 ½ pounds of chocolate is put in ¾ pound boxes. How many full boxes are made? How much chocolate is left? 5 ¾ purdinkles are split into buckets that hold 2/3 of a purdinkle. How many buckets will we need? (You don’t need to know what a purdinkle is.) 2 ¾ yards of fabric are needed for each shirt. How many yards for 10 shirts? How many shirts could be made from 30 yards? Tiffany splits 4/5 pounds of nails into 4 piles. How many pounds in each pile? A cube (box with all sides the same length) is 3 ½ inches on each side. What is the volume of the cube? Hint: Multiply all three sides. (3 ½)3 Find the surface area of the cube above. Hint: 6 times (3 ½ )2 How many cubes can be made of a piece of metal 367 ½ square inches? How much metal is left over? i) j) k) Write a word problem that requires multiplication of fractions then subtraction to solve. 3 2-6-2 Word Problems a) If one yard of polar fleece fabric costs $12.00, how much does 5 ½ yards cost? One cubic foot of water weighs 62 ½ pounds. How much does 5 3/8 cubic feet of water weigh? My mileage gauge reads 4533 3/10 miles. (Ok, I admit it has rolled over.) Last week it read 4042 7/10 miles. How far have I driven this week? b) Janet worked with students for 5 ¾ hour on Tuesday, 4 3/8 of an hours on Wednesday, and 1 2/3 hours Thursday. How much time did she spend with students for the three days? c) The shopping list included 3 pounds of hamburger, 3 ½ pounds of apples, 2 ¾ pounds of oranges, and 1/3 pound kiwi. How many pounds of fruit were purchased? The baby was 6 2/5 pounds at birth two months ago. He gained 4 ½ pounds. How much does he weigh now? d) How many pieces of wire each 3 ¾ inches long can be cut from a roll that is 50 inches long? One slide is 5 3/8 feet taller than the other. The short slide is 12 ¾ feet. How tall is the tall slide? e) The records showed 123 ¾ pounds of monkey food were still in the inventory. Each day 25 2/3 pounds are used. How much is left after three days? Monster costumes for the play each required 5 3/5 yards of fabric. How many yards will be needed to make all 12 costumes? f) How many quart jars of jam can be made from 34 5/8 pound of strawberries? 2/3 of a pound will make one quart jar. How many 1 2/3 pound boxes of cookies can be made from 10 ¾ pounds of cookies? g) Jamie spent 3 ¾ hours typing word problems, 2 ½ hours making answer keys, 2/3 of an hour making copies. One twelfth of her time was spent yelling at the computer. The rest of her eight hour day was spent filing. How much time was spent filing? One section of a wall of my house will be a book shelf. The section is 6 ½ feet long and 10 feet high. Most shelves should be 1 foot tall, but the two bottom shelves will be 1 ¾ feet tall for oversized books. This will leave the top shelf an odd height. How much shelving is needed? Don’t put shelving on the ceiling, but the floor will need shelving. 4 h) While doing home health visits, Jack drove 3 ½ miles to Mrs. Jenkins, 5 7/10 miles to the next appointment, 4 2/5 to Mr. Smith, and 7 miles back to the office. What should he claim on his mileage report? Jon will work overtime for time and a half. He works a 10 hour day. His regular pay is 12.50 per hour. How much did he make? i) The family got three 2 ½ pound boxes of chocolate for Christmas. How much chocolate does each of the five people get? Larry was home from work because he was ill for 5/12 of his 40 hour work week. How many hours did he work? j) Jon's full truck weighs 3 ¾ tons. The average crate weight 2/5 ton. If there are seven crates on the truck, how much does the empty truck weigh? If you change the rotating speed of a cement mixer from 12 rpm's to 5 1/4 rpm's how many revolutions per minute slower does the cement mixer spin? k) Jackie lost 10 3/5 lbs. Before the diet she weighed 140 ½ lbs. How much does she weigh now? Rosy cut 3 ¾ feet from a 10 foot piece of sheet metal. How long was the remaining piece? l) Zuella made some dresses out of 15 ¾ yards of fabric. Each required 3 2/3 yards. How many dresses was she able to make? I only have ¾ pounds of German baking chocolate left. My recipe calls for 5/8 pounds. How much will I have left? m) A ten pound bag of cotton balls has been opened. (It's a really big bag.) Yesterday there was 8 2/3 pounds left. If someone uses 7/8 of a pound, how much do the remaining cotton balls weigh? Todd worked 9 ½ hours Monday, 10 ¾ hours on Tuesday, 8 hours on Wednesday, 8 hours on Thursday, and 9 hours on Friday. He makes $14 per hour regular pay and time and a half for anything over 40 hours. How much did he make that week? The trip to the store is 7/8 miles. Joan walked 2/3 of the way. How far has she walked? n) Craft trim is to be cut in 5 3/8 in. pieces. How many pieces can be cut from 3 ½ yards? 5 2-6-3 Exponent Rules Remember: 34 3(3)(3)(3) so 34 33 3(3)(3)(3)(3)(3)(3) 37 x5 x3 xxxxxxxx x8 Rule: a a a m n m n and so 23 s 2 32 s5 8ss9sssss 72sssssss 72s 7 f 52 f 123 f 175 1 3 7 and 2w 8 3w 4 6w 8 (don’t worry what fractional exponents mean…yet.) When multiplying like bases, add the exponents. Practice: Simplify. a) a1a1 = b) c) d) e) f) g) h) i) j) 3 y 3 z 4 y 5 z 5 12 y 35 z15 12 y 8 z 6 e2e2 c 2c7 = d 5d 1 = c 25c 33 = d 5 d 1 = 8c 2 5c 7 = 3d 5 9d 2 = 4 2 c 3 52 c 7 = 5 2 d 5 4d 1 = 4 x 3 c 5 x 9 10c 7 = x 3.4 d 5 x 3 d 2 5 = g9g7 = h5h5 = g 29 g 47 = b 6b3 f 4 f 8= e 71e 53 a10 a10 = b 600b 30 4a 1 4a 1 = 4b 7 7b 4 25 y 3 32 a = f N f N= 2e 2 2e 2 34 f 3 4 3 f 8 = Go M Et s h 3h 2 = Ng 9 Ng 7 = 3(31) h 7 34 h 5 = 44 g 4 44 g 4 = 5 2 h 5 55 h 5 = 6 5 36 b 6 b 3 3a1 x 2 a 2 x1 3 = x 5 2b 5 2 x 3b 3 43 f 4 63 f 8 = (9 1) xe 4 x10e 4 7x6 f 4 x9 f 9 = 1.5 x1.1 g 9 7.5 x 2.1 g 6 = 4 x 6 h 7 x9h 2 = Remember how to cancel. Make a 1 out of common factors. 14 6 7 2 7 4 15 2 5 5 r 5 rrrrr rrrrr rr h3 hhh 1 1 r2 3 r3 rrr rrr 1 h6 hhhhhh hhh h a 3b 2 aaabb aaabb a a 2 2 4 ab aabbbb aabbbb bb b am mn Rule: n a a Review the examples. k 543 k 543215 k 328 215 k is the same as 12s 25 3 9 s 20s16 5 When dividing like bases, subtract the exponents. r rrrrr rrrrr rr r2 3 r rrr rrr 1 5 r 53 6 Practice: Simplify. a) w4 b) c) d) e) w2 q5 q4 q 24 q10 4e7 12e2 2 x3 5 x5 15 x 4 e5 e3 t8 t3 x 54 x110 14 f 14 21 f 12 15 y13 21y 52 32 x 45 r8 r2 w4 w4 m 45 m73 45 x 64 9 x54 4a 25 5a8 20a30 y7 y5 x 21 x 21 w84 w92 144k 12 60k 5 12a5 56a81 64a50 p12 p8 b14 b13 f 31 f 72h46 56h45 3x3 2 x 2 5 x5 Notice what happens with the example This leads to another rule. h3 hhh 1 1 3 h36 h 3 6 h hhhhhh hhh h Rule: a n s 3 1 s3 and 1 d5 5 d 1 1 and n a n an a Practice: Write each answer two ways, as a fraction with positive exponents and without using fractions but using negative exponents. f) 1 x w4 w32 w-3 or 3 5 7 34 g) w a7 a12 w x q 41 q 52 w s4 s5 w4 w2 z14 z15 36r 2r 5 s12 9 4 2rr 5 s12 8r 6 s12 8 67 127 7 7 r s 45r 7 s 7 9 5r 7 s 7 5r s 5 h) = 8s 5 5r = 8 1 5 r s 5 i) 3e3 4e6 30 j 30 25 j 60 7f7 4f6 j3 5 j10 81m3n5 144m3n7 9 p3 3 p5 12t 8 15t 15 35s 2t 5 21s 3t 17 36 y13 6 y16 56r 98 48r 123 32 x51 y15 12 x30 y 71 j) 28 x5 y 5 21x3 y 7 k) 15m3 3m3 45m6 25m5 33m5 55m10 6 p1310 p13 8 p16 15t 3 60t 3 45t 7 7 l) 32t16 56t 25 10k 12l 51 45k 36l 27 3m14 3m13 45m61 2t 6t k 31l 5k 36l 7 42e65 48e79 30k l 52 10k 36l 3 12t 90 48t 205 m) 24 f 12e41 4e26 f 27 n) x3 x33 x0 3 x or all cancel for an answer of 1. Rule: a 1 0 Anything to the zero power is 1. Practice: Simplify o) 30= x0= 0 p) 4x = 8e0= (4y)0= 2s5x0= (3st5)0= 2(3st5)0 (5-8+9-7-12)0= ½ (3st5)0 Rule: a m n a mn x 3 5 x3 x3 x3 x3 x3 x35 x15 using a previous rule. An easier way would be to multiply the 3 and 5 to get the new exponent of 15. Practice: Simplify. q) 2 3 s x 12 3 4 w 4 5 2 x u 3x 4a 21 3 5 2 r) s) t) x 2x 4a b c 21 3 2 3 2 5 y 5r 64a6b15c3 5 5 25 4 5 4 m 2 3 3 5 52 6 3t 2 5 2 5 12 2 3 3d e f = 2u v u 5 2b51 u) 2 2 5 2 = 2.5 2 3 = 0.2x 0.5x 3 2 0.5r 1.2r 3 4 3 v) 3 xy 3 6 x5 y 2 2 = x5 y 3 x5 y 2 3 = 56kl 310k 3 16k 8l 214k 10 2 = 12a 3 4a 5b3 6a8b5 2 = 8 5x3 3x2 are not like terms and cannot be combined. 3 4 7 Contrast this with 5x 3x 15x The multiplication can be done. 4m3 5m3 9m3 Don’t add the exponents. 4m3 5m3 20m6 Do add the exponents. a) 3d 5 2d 5 3d 5 2d 5 2 x3 2 x5 Notice: b) c) d) 4s-7s+8s2= 2 x3 2 x5 3u3 2u5 7r4(7r3)= 7e5 2e5 7e5 2e5 4s(-7s)(8s2)= 3x2(-4x2)(-8x2)= 3u3 2u5 7r4+7r3= w-6w-(-7w2)= 3x2-(-4x2)-8x2= w(-6w)(-7w2)= Mixed practice: Simplify e) 4q 3 p18 5q 8 p 9 = f) g) 4q3 p18 5q 8 p 9 4q 3 p18 = 2 3t 3 5t 2 4 3 2 3 a 3 4 a 512 a 36 x8 y 3 48 x 7 y12 4 3nm 2 2 5nm2 2nm2 = 4e2 7e3 3e = = 3a 2 1 3 a 5 3 2w 3 2 7 w3w5 34w3 h) i) 3x2 4x2 7 x2 = 2 5 4e 7e 3e 2 3 3s 5s 5 3 4 y6 z 67 y3 12 y 5 z 4 15 t 2 38 t 3 2 3 t 3 5 x 2 x 4 y5 x 3 5 10 xy 7h 5 18h9 2 3 5 h2 j) 3 8 3 8 t 2 38 t 3 2 3 t 3 3 10 x3 x9 3 9 2-6-4 Scientific Notation Scientific notation is used to write very large and very small numbers. The distance to the sun is150, 000,000km or 1.5x108km. The bacteria Streptococcus pyogenes is 0.0000625 cm or 6.25x10-5cm long. 4x103=4x10x10x10=4x1000=4000 Notice: the exponent on the 10 is 3 and the decimal after the 4 moved to the right 3 spaces. 2.3x105 = 23000 moving the decimal right 5 spaces. 2.3x105 is scientific notation and 23000 is decimal notation. Practice: Write the following in decimal notation. a) 4.6x108= b) c) 7x103= 1.123x109= m 5.2x103= 8X108= 5.453x104= 3.4x1012= 1x1010= 6.5874x103= Rule: a 10 is scientific notation if 1 a 10 and m is an integer. 2.3x108 = 23x107 = 0.23x109 = 230x106 But only one of these is in scientific notation. 2.3 is between 1 and 10 so the correct form of the number in scientific notation is 2.3x108. Although the other numbers are the same as 2.3x108, they are not scientific notation. 3.21x102 is the scientific form of 321. To write 45600000 in scientific notation, 4.56x107 = 45600000 move the decimal so only one digit is in front and drop ending zeros. 4.56 Multiply by ten to the power that matches the number of spaces the decimal was moved. Practice: Write the following in scientific notation. a) 8,900,000,000,000= 2,000,000,000,000,000= b) c) 1,000,000,000,000= 8,431,000,000,000,000= 4 178,000,000,000,000= 145,000,000,000,000,000= 5,000,000,000,000= 24,000,000,000,000= 1,000,000,000,000,000,000,000,000= Recall 10 1 1 6.8 4 so 6.8 10 = 0.00068 Dividing by 10000 moves the decimal 4 10 10000 10000 4 spaces to the left. To change a small number in scientific notation to decimal notation, move the decimal to the left the number of spaces indicated by the power of 10. 5.43x10-3 = 0.00543 (left) and 6.43x103= 6430 (right) Practice: Write the following in decimal notation. Watch the signs. a) 4.6x10-8= 5.2x10-3= b) 7x10-3= 1.8X10-8= 3.4x10-12= 1x10-10= 10 c) d) e) 1.123x10-9= 4.8x103= 3x10-5= 5.453x10-5= 2.3x10-5= 3.3x10-6= 6.5874x10-3= 4.5x10-8= 8.1x10-9= To write a small decimal notation number to scientific notation, move the decimal after the first nonzero digit. Count how many spaces the decimal was moved. This is the power of 10 the number would be divided by. Make it a negative exponent. 0.00003123 = 0.00003 123 3.123 10 5 Practice: Write the following in scientific notation. Watch the signs. a) 0.000000043= 0.00000254= b) 0.0000235= 0.00007= c) 0.000000000028= 0.00000000000009= Practice: Write the missing numbers. Decimal Notation d) e) f) g) h) i) The mass of the moon is 73,500,000,000,000,000,000,000. The diameter of an artery is .00621 inches. The population 6230,000,000. of the world is about 0.000023= 0.0000000001= 0.00001= Scientific Notation The population of the United States is 3x108. The human heart is 3.1x100 inches in diameter. The moon is 3.8x105 km away. j) k) The mass of the Earth is 5.97x1024 kg. A hair grows 2.08x10-4 mm per minute. 11 Multiplication and Scientific Notation To multiply 3.1 x 104 X 2.3x105, use the commutative property 3.1x2.3x104x105 and the associative property (grouping of multiplication doesn’t matter) to get 7.13x104x105. Then use the exponent rules to get the final result 7.13x109. (3x108)(1.5x1023)=4.5x1031 (2.1x104) (4x10-12) =8.4x10-8 (4.6x108) (5.8x106) =26.68x1014 Notice: What is wrong with the last example? Although the answer is correct, the number is not in scientific notation. To finish the problem, move the decimal one space left and increase the exponent by one. 26.68x1014 = 2.668x1015 If the decimal moves right, the exponent moves down. 0.00042x109=4.2x105 If the decimal moves left, the exponent moves up. 7890x103=7.89x106 Practice: Multiply then write the following in scientific notation. a) 3 105 2.3 109 2.1105 1.3 109 2 10 10 23 7.2 10 110 5 9 b) c) 1.2 10 5.3 10 15 15 3.4 10 2.3 10 51 3 10 2.3 10 5 9 2 10 3 10 9 9 The negative in front of the 2 makes a number less than zero. The negative in front of the exponent makes a small number. 2.5 10 3 10 5 12 The following will need to have the decimal moved and the exponent adjusted after the multiplication. d) 5.4 107 3.1108 3.25 109 5.3 1012 2 1012 7.3 1023 e) f) 5.48 10 3.2 10 15 10 7.2 10 4.3 10 21 35 2.5 10 8.3 10 4 8 The population of the world is 6.23x109 and the average person consumes 5x103 grams of sugar per year. How many grams of sugar are consumed in the world per year? 12 Division and Scientific Notation 4.5 108 4.5 Division works in a similar way. 1085 1.5 103 5 3 10 3 1.092 1013 1.092 1013( 5) 0.52 108 5.2 109 Notice the decimal and exponent. 5 2.1 10 2.1 6.8 103 3 8 6.8 10 3.4 10 3.4 108 2 1038 2 105 Practice: Divide the following then write the answer in scientific notation. a) 1.5 1021 5 1071 b) c) 2.3 105 1103 1108 2.3 1052 1103 3.4 108 8.3 107 1.2 105 6.8 103 3.4 108 A certain string of bacteria is 3.4x10-5 cm long. There are approximately 80 cells in a string How wide is each single bacteria? Calculator notation Some calculators show scientific notation the same way we have written them here. Some use EE. They write 3.456 EE5 to mean 3.456 x105. Some write the same thing with this in the screen 3.45605. There are several ways electronic equipment displays scientific notation. Look in your user’s manual to see how it is displayed on your calculator. 13 2-6-5 Roots 9 3 because 3 times 3 is 9 25 5 because 5 times 5 is 25 When working a square root problem, ask: “What times itself is the number inside the root symbol?” 27 3 because 3x3x3 is 27. The small 3 outside the root symbol tells how many times the answer must be multiplied to get the number inside the root. 3 5 32 2 because 2x2x2x2x2 or 25=32 125 1024 243 4 5 Practice: Simplify. a) 3 64 b) c) d) 4 4 3 3 5 5 3 6 216 64 81 625 100000 4 8 16 256 Prime factorization is writing a number using multiplication of only prime numbers. 12 can be written as 3x4 ,but 4 is not prime and can be written as 2x2, so the prime factorization of 12 = 3x2x2 This can be written 3x22. To write 330 using its prime factorization, start breaking it up into smaller factors until there are no more composite numbers. 330=33x10=3x11x2x5 Practice: Write the prime factorization for the following. a) 12 8 Prime number: a number with only factor of 1 and itself. 2 3, 5, 7,11,13,17, 19, 23... are prime numbers. 15 is not prime because 3 and 5 also divide it evenly. 15 is a composite number. 330 10 5 x 2 x x 33 3 x 11 100 27 b) 56 28 36 32 c) 50 42 18 25 d) 75 48 360 72 e) 7875 50176 7200 3136 f) 405 432 5488 9375 g) 864 442368 3600 1225 14 Simplifying roots. You won’t be using the root button on your calculator for these. 45 3 3 5 32 5 3 5 4 First write the number as it’s prime factorization. Then, because this was a square root, a pair of 3s can be simplified to a 3 outside the root. To get an exact answer, leave the 5 inside the root rather than using a calculator. 4 19440 Prime factorization 4 2 2 2 2 3 3 3 3 3 5 four 2’s and four 3’s and fourth root so 2 4 4 34 4 3 5 2 34 15 64 15 This is the simplest form of the root. 245 360 968 18 Practice: Write the roots in simplest form. a) 75 5 5 3 5 3 b) c) d) e) f) 3 28 392 2700 300 3 2 2 5 5 2 5 3 10 3 576 2016 500 3 5 5 5 2 2 53 4 400 4 2 2 2 2 5 5 24 25 3 1125 3600 70000 3 3528 1080 567 1575 3 576 2744 4 4 4 20000 4 1296 Using a calculator will give an approximation. In the example above 4 19440 11.81 rounded to the nearest hundredth. To use your calculator, you need to learn another notation. Rule: x x 1 Example: 4 52 52 4 520 25 On your calculator find the xy button or yx. Type 52 then hit the yx button. Then type .25 = 2.69 (Rounded) 1 Practice: First, rewrite the root using a fractional exponent. Then use your calculator to find the answer rounded to the nearest hundredth. 10 5 g) 4 123 1231 / 4 type 123 yx then .25 = 100000 159 h) i) j) 3 25 25 3 1 3 type 25 yx then (1/3)= 4 578 52 5 3 2 3 951 4 53 5 4 5.75 20 3 3 85 345 48 All the rules that work for exponents work for these fractional exponents. This is learned and practiced in future math classes. 15