UNIT 3: POWERS. SQUARE ROOTS. SCIENTIFIC NOTATION. PERCENTAGES. 3.1. POWERS 3.1.1. POWERS OF INTEGERS A power is an abbreviated way of writing a product of equal factors. a a a a a = a in powers, the repeated number is called base and the number of times that is repeated is called index. Important: If the base is a positive number, the power is always positive. If the base is a negative number, the power will be positive if the index is an even number, and the power will be negative if the index is an odd number. Examples: Ø 2 = 2 2 2 = 8 Ø 2 ( = 2 2 2 2 = 16 Ø 4 = 4 4 = 16 Ø 3 = 3 3 3 = 27 3.1.2. POWERS OF FRACTIONS To raise a fraction to a power, we rise the numerator and the denominator to that power. For expample: 3 5 = 3 5 = 27 125 As we said with powers of integers, when we raise a positive fraction, the power will be positive; when we raise a negative fraction, the power will be positive if the index is an even number, and negative if the index is an odd number. 3.1.3. OPERATIONS WITH POWERS Product of powers with the same base To multiply powers of the same base, we keep the base and add the indexes. a 1 a 2 = a 132 1
Quotient o powers with the same base To divide powers of the same base, we keep the base and subtract the indexes. a 1 : a 2 = a 152 Power of another power To raise a power to another power, we keep the same base and multiplying the indexes. a 1 2 = a 1 2 Power of a product (product of powers with the same index) The power of a product is equal to the product of the powers of the factors. a b 2 = a 2 b 2 Power of a quotient (quotient of powers with the same index) The power of a quotient is equal to the quotient of the powers of the factors. a: b 2 = a 2 : b 2 a b 2 = a2 b 2 3.2. SQUARE ROOTS 3.2.1. SQUARE ROOT OF AN INTEGER The square root of a number a, is another number b that when it is raised to the power of two, we get the number a. a = b b 2 = a The " " symbol is called the "radical" symbol. The expression " 9 " is read as "root nine", "radical nine", or "the square root of nine". Note: Numbers with an exact square root are called perfect squares. A positive integer has always two square roots, one of them is positive and the other is negative. A negative integer does not have any square root because when we raise to the power of two, we always get a positive number. 3.2.2. SQUARE ROOT OF A FRACTION The square root of a fraction is the quotient between the square root of the numerator and the denominator. < = = < = As we said in integers, a positive fraction has two square roots, the positive and the negative. 2
A negative fraction does not have any square root. 3.3. SCIENTIFIC NOTATION 3.3.1. POWERS WITH BASE 10 The value of a power with base 10 is an 1 followed by so many 0 as the index says. For example: 10 5 =100 000 3.3.2. EXPRESSION OF BIG NUMBERS We usually use the scientific notation to express very big or very small numbers. A number is written using scientific notation when it is expressed as the product of a number between 1 and 9 and a power with base 10. The index of the power with base 10 is called order of magnitude. Example: the extension of Europe is 10530000 km = 1,053 10 C 3.4. PERCENTAGES A percent is a ratio of a number to 100. A percent is expressed using the symbol %. A percent is also equivalent to a fraction with denominator 100. To calculate the percentage of a quantity we must multiply it by the percent and divide by 100. Example: 5% of 72 is C DEE = 3,6 Calculate the number when we know the percentage We must multiply the percentage by 100 and divide by the percent. Example: The 22% of a number is 66, which is the number? Total Part 100 66 = 22x 100% 22% x = FF DEE = 300 x 66 Calculate the percentage when we know the total quantity and the part of it We are going to study that with an example: 1200 interviews have been made in a school. 876 students have answered they clean their teeth every day. What is the percetage of students that clean their teeth every day? Total Part 100% x% 3
1200 876 Calculate a number increased with a percentage We add to the number the percentage. Example: Calculate the value of 320 increased by a 5% First step: the increase is 5% of 320 is E DEE = 16 Second step: the final value is 320 + 16 = 336 Calculate a number decreased with a percentage We subtract to the number the percentage. Example: Calculate the value of 320 decreased by a 12% First step: the decrease is 12% of 320 is D E DEE = 38,4 Second step: the final value is 320 38.4 = 281.6 EXERCISES AND PROBLEMS 1. What sign will have the following powers? a) 3 C b) 3 c) G 5 F d) H 2. Calculate: a) 2 ( = b) 2 = c) 1 C = d) 5 = e) 7 = f) 9 = g) Ḏ F = h) = 4
i) F = j) DD 5 = 3. Apply the properties of the powers to get only one power. a) 2 2 F b) 3 H E c) 2 3 d) D ( : D e) f) 5 ( g) DE ( h) 13 2 C i) 9 : C : C j) k) 15 F : 5 F 4. Apply the properties of the powers and calculate the result: a) ( b) Ḏ : Ḏ ( 5. Write the following number in scientific notation. a) 475 000 b) 324 000 000 c) 800 500 000 6. Complete: a) 89 000 =... 10 ( b) 30 500 = 3,05 10. 7. Write these numbers with all the figures: a) 1,5 10 ( 5
b) 9,03 10 c) 5,607 10 G d) 6,01 10 DD 8. Calculate and express the result with scientific notation. a) The distance in kilometers that the light goes through one year (speed of the light: 300000 km/s) b) The mass in kilograms of 500000litres of mercury (1 litre of mercury weighs 13,6 kg) 9. In a school, 25 % of the teachers teach basic maths. If there are 50 basic maths teachers, how many teachers are there in the school? 10. There are 60 students in a class. 85% of them study French. How many students do not study French? 11. A football team won 65% of the matches last year. If they won 26 matches, how many matches did they play in total? 12. Laura earns 26000 a year and she pays 5200 in taxes. Esteban earns 46500 a year and he pays 8370 in taxes. Who pays a higher percentage of taxes? 13. A compact disc priced at 12 is on sale at a 20% discount. How much does the disc cost on sale? 14. In 2014, a car insurance cost 300. In 2015, it decreased by 5%. In 2016 it increased by 10%. What is the final cost of the car insurance? 15. A bike was sold for 2760 in January with a 8% discount. How much did it cost before the discount? 16. Sonia bought a book for 20. The cashier discounted it by 2% because the book was on sale, and after that, he added a 4% IVA. What is the final price of the book? 17. 40% of the people in a neighbourhood are older than 65 years old. How many people are older than 65 years old? 6
18. In the last elections 4560 people voted, and that represents the 75% of the total people in the town. How many people are there in the town? 19. Teresa bought 20 books at the price of 15 euros each book. She got a 10% discount in each one. How much money did she spend? 20. A trader used to sell coffee at the price of 5 /kg. Now he is selling it at the price of 4,75 /kg. What is the discount percentage? 21. There are 1200 students in a school. 25% play tennis, 15% play basketball and 40% play football. Calculate the number of students that play each sport and the percentage of the students that do not practice any sport. 22. Three mountaineers are carrying food for a trip. When they arrive to the shelter, they find out that they have increased their provisions by 15% because the have gathered fruit from the trees. If they now have 402,5 kg of food, calculate how many kilograms of food they had at the beginning of their trip. 23. Last year the reserve of water in a country was 350 hm3. If this year the amount of water has increased by 12%, how many hectometers of water are there now? 24. In a class of 30 students there are 30 people who have the flu. What is the percentage of students who have the flu? 25. We want to make a photocopy reducing the height of the photocopy from 12,5 cm to 6 cm. What percentage of reduction are we applying? 26. The price of a product was increased by 20% in January and another 5% in February. The final price was 504. What was the initial price? 27. The price of a fridge with 21% IVA (VAT) is 605. What is the price without IVA? 28. The price of a jumper was 38 but I got a 25% discount. If we have to apply a 21% IVA (VAT) over that price, what would the final price would be? 7
29. A trader decided to increase the price of merchandise which costs 72 by 3%. If next week he decided to increase the price another 3%, what will the final price be? 30. Helen makes lemonade with 12 litres of water and 8 litres of lemon juice. What is the percentage of lemon juice? 8