4 ESO Academics - UNIT 02 - POWERS, ROOTS AND LOGARITHMS
- 1. Unit 02 October 1. INTEGER EXPONENT POWERS. 1.1. INDEX OR EXPONENT NOTATION. Instead of writing 2 𝑥𝑥 2 𝑥𝑥 2 𝑥𝑥 2 𝑥𝑥 2 we can write 25 : In 25 , the 2 is called the base number and the 5 is the index, power or exponent. The index is the number of times the base number appears in the product. This notation enables us to quickly write long lists of identical numbers being multiplied together. 34 is the short way of writing 3 𝑥𝑥 3 𝑥𝑥 3 𝑥𝑥 3 106 is the short way of writing 1,000,000 = 10𝑥𝑥10𝑥𝑥10𝑥𝑥10𝑥𝑥10𝑥𝑥10 MATH VOCABULARY: Base Number, Index, Exponent. Distribution. 1.2. NAMING POWERS. 65 may be read as: • Six to the fifth power • Six to the power of five • Six powered to five. NOTE: During this curse use the first way. Axel Cotón Gutiérrez Mathematics 4º ESO 4.2.1
- 2. Unit 02 October 1.3. SQUARE AND CUBE POWERS. We call Square Power a number to the second power. It represent the result of multiplying a number by itself. The verb "to square" is used to denote this operation. Squaring is the same as raising to the power 2. We call Cube Power a number to the third power. It represent the result of the number multiplied by itself twice. The verb "to cube" is used to denote this operation. Cubing is the same as raising to the power 3. MATH VOCABULARY: Square, Cube, Raise, To Factorize. Axel Cotón Gutiérrez Mathematics 4º ESO 4.2.2
- 3. Unit 02 October 1.4. INTEGER EXPONENT POWERS. If 𝒂𝒂 ∈ andℝ 𝐧𝐧 ∈ ℕ ⇒ 𝐚𝐚𝐧𝐧 = 𝐚𝐚 ⋅ … ⋅ 𝐚𝐚����� 𝐧𝐧 𝐯𝐯𝐯𝐯𝐯𝐯𝐯𝐯𝐯𝐯 𝒂𝒂−𝒏𝒏 = 𝟏𝟏 𝒂𝒂𝒏𝒏 = 𝟏𝟏 𝐚𝐚 ⋅ …⋅ 𝐚𝐚����� 𝐧𝐧 𝐯𝐯𝐯𝐯𝐯𝐯𝐯𝐯𝐯𝐯 ; 𝒂𝒂 ≠ 𝟎𝟎 Moreover, ∀𝐚𝐚 ≠ 𝟎𝟎; 𝐛𝐛 ≠ 𝟎𝟎 ⇒ 𝐚𝐚𝟎𝟎 = 𝟏𝟏 𝐚𝐚𝟏𝟏 = 𝐚𝐚 𝐚𝐚−𝟏𝟏 = 𝟏𝟏 𝐚𝐚 � 𝐚𝐚 𝐛𝐛 � −𝟏𝟏 = 𝐛𝐛 𝐚𝐚 23 = 2 ∙ 2 ∙ 2 = 8 2−3 = 1 2 ∙ 2 ∙ 2 = 1 8 (−5)−3 = 1 (−5) ∙ (−5) ∙ (−5) = − 1 125 � 2 −3 � −1 = −3 2 = − 3 2 1.5. PROPERTIES OF POWERS. When powers with the same base are multiplied, the base remains unchanged and the exponents are added. 𝒂𝒂𝒏𝒏 ∙ 𝒂𝒂 𝒎𝒎 = 𝒂𝒂𝒏𝒏+𝒎𝒎 75 ∙ 73 = (7 ∙ 7 ∙ 7 ∙ 7 ∙ 7) ∙ (7 ∙ 7 ∙ 7) = 75+3 = 78 Axel Cotón Gutiérrez Mathematics 4º ESO 4.2.3
- 4. Unit 02 October When we have powers with different base but the same exponent, we multiply the bases and keep the same exponent. We may also do it in the reciprocal way. ( 𝒂𝒂 ∙ 𝒃𝒃)𝒏𝒏 = 𝒂𝒂𝒏𝒏 ∙ 𝒃𝒃𝒏𝒏 53 ∙ 73 = (5 ∙ 5 ∙ 5) ∙ (7 ∙ 7 ∙ 7) = (5 ∙ 7) ∙ (5 ∙ 7) ∙ (5 ∙ 7) = (5 ∙ 7)3 = 353 64 = (3 ∙ 2)4 = 34 ∙ 24 When powers with the same base are divided, the base remains unchanged and the exponents are subtracted. 𝒂𝒂𝒏𝒏 ÷ 𝒂𝒂 𝒎𝒎 = 𝒂𝒂𝒏𝒏 𝒂𝒂 𝒎𝒎 = 𝒂𝒂𝒏𝒏−𝒎𝒎 75 ÷ 73 = (7 ∙ 7 ∙ 7 ∙ 7 ∙ 7) ÷ (7 ∙ 7 ∙ 7) = 7 ∙ 7 ∙ 7 ∙ 7 ∙ 7 7 ∙ 7 ∙ 7 = 75−3 = 72 When we have powers with different base but the same exponent, we divide the bases and keep the same exponent. We may also do it in the reciprocal way. ( 𝒂𝒂 ÷ 𝒃𝒃)𝒏𝒏 = � 𝒂𝒂 𝒃𝒃 � 𝒏𝒏 = 𝒂𝒂𝒏𝒏 𝒃𝒃𝒏𝒏 = 𝒂𝒂𝒏𝒏 ÷ 𝒃𝒃𝒏𝒏 153 ÷ 33 = (15 ∙ 15 ∙ 15) ÷ (3 ∙ 3 ∙ 3) = (15 ÷ 3) ∙ (15 ÷ 3) ∙ (15 ÷ 3) = (15 ÷ 3)3 = � 15 3 � 3 = 53 When we have a power of powers, the exponents must be multiplied: ( 𝒂𝒂𝒏𝒏) 𝒎𝒎 = 𝒂𝒂𝒏𝒏∙𝒎𝒎 (23)5 = 23 ∙ 23 ∙ 23 ∙ 23 ∙ 23 = 23+3+3+3+3 = 23∙5 = 215 Axel Cotón Gutiérrez Mathematics 4º ESO 4.2.4
- 5. Unit 02 October 2. ROOTS AND RADICALS. The nth root of a number is written as𝐚𝐚 √𝐚𝐚, called 𝐧𝐧 radical, and is the number that must be multiplied by itself n times to equal the number .𝐚𝐚 √729 3 = 9 ⇔ 93 = 729 √−343 3 = −7 ⇔ (−7)3 = −343 We have different types of radicals: √ 𝒂𝒂 𝒏𝒏 Radicand Index Number of Roots 𝒂𝒂 > 𝟎𝟎 n odd 1 root: positive n even 2 roots: 1 positive and its opposite 𝒂𝒂 = 𝟎𝟎 n odd or even 1 root: √𝟎𝟎 𝒏𝒏 = 𝟎𝟎 𝒂𝒂 < 𝟎𝟎 n odd 1 root: negative n even no Real root MATH VOCABULARY: Root, Radical, Radicand, Index. Axel Cotón Gutiérrez Mathematics 4º ESO 4.2.5
- 6. Unit 02 October 3. FRACTIONAL EXPONENTS. Radical expressions can be rewritten using fractional exponents, so radicals can be expressed as powers. √𝐚𝐚 𝐧𝐧 = 𝐚𝐚 𝟏𝟏 𝐧𝐧 ⇔ �𝐚𝐚 𝟏𝟏 𝐧𝐧� 𝐧𝐧 = 𝐚𝐚 𝐧𝐧 𝐧𝐧 = 𝐚𝐚𝟏𝟏 = 𝐚𝐚 √ 𝒂𝒂 𝒎𝒎𝐧𝐧 = 𝐚𝐚 𝒎𝒎 𝐧𝐧 ⇔ �𝐚𝐚 𝒎𝒎 𝐧𝐧 � 𝐧𝐧 = 𝐚𝐚 𝐦𝐦∙𝐧𝐧 𝐧𝐧 = 𝐚𝐚𝐦𝐦 �√729 6 � 2 = ��936 � 2 = �9 3 6� 2 = �9 1 2� 2 = 9 2 2 = 91 = 9 Two radicals are equivalents when both can be expressed as fractional exponent powers, with the same base and equivalent index. 𝐚𝐚 𝐦𝐦 𝐧𝐧 𝐢𝐢𝐢𝐢 𝐞𝐞𝐞𝐞𝐞𝐞𝐞𝐞𝐞𝐞𝐞𝐞𝐞𝐞𝐞𝐞𝐞𝐞𝐭𝐭 𝐭𝐭𝐭𝐭 𝐛𝐛 𝐩𝐩 𝐪𝐪 ⇔ 𝐚𝐚 = 𝐛𝐛 𝐚𝐚𝐚𝐚𝐚𝐚 𝐦𝐦 𝐧𝐧 = 𝐩𝐩 𝐪𝐪 2 5 4 𝑖𝑖𝑖𝑖 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 𝑡𝑡𝑡𝑡 2 10 8 Since we can represent radicals as powers, all the properties of powers are applied to radicals. MATH VOCABULARY: Equivalent Radicals. Axel Cotón Gutiérrez Mathematics 4º ESO 4.2.6
- 7. Unit 02 October 4. OPERATION WITH RADICALS. 4.1. REDUCTION OF RADICALS TO SAME INDEX. To reduce radicals to the same index we have to express them as fractional exponent powers and look for the same denominator using the lowest common multiple (LCM). Reduce to the same index: √5, √743 : √5 = 5 1 2; �743 = 7 4 3 We have to do the LCM of the exponent: 𝐿𝐿𝐿𝐿𝐿𝐿(2,3) = 6 ⇒ 1 2 = 3 6 𝑎𝑎𝑎𝑎𝑎𝑎 4 3 = 8 6 Therefore: 5 1 2 = 5 3 6 = �536 𝑎𝑎𝑎𝑎𝑎𝑎 7 3 4 = 7 8 6 = �786 MATH VOCABULARY: Lowest Common Multiple (LCM), Highest Common Factor (HCF). 4.2. SIMPLIFYING RADICALS. You can simplify radicals by expressing them as fractional exponent powers and obtaining the simplest fraction of the exponent. Simplify √3525 �3525 = 3 5 25 = 3 1 5 = √3 5 Axel Cotón Gutiérrez Mathematics 4º ESO 4.2.7
- 8. Unit 02 October MATH VOCABULARY: Simplest Fraction. 4.3. EXTRACTING FACTORS. To extract factors from a radical, we have to factorize the radicand and express its factors as powers. Then we have to extract the factors which exponent is higher or equal than the radical index by using the powers properties. Example 1: �3,888 3 3,888 = 24 ∙ 35 ⇒ �3,888 3 = �24 ∙ 353 The factor exponents (4 and 5) are greater than radical index (3), so we can at least extract one of those factors using power properties: �24 ∙ 353 = (24 ∙ 35) 1 3 = 2 4∙1 3 ∙ 3 5∙1 3 = 2 4 3 ∙ 3 5 3 Now we use the product power property to convert the improper fraction in a integer plus a proper fraction 2 4 3 ∙ 3 5 3 = 2 3 3 ∙ 2 1 3 ∙ 3 3 3 ∙ 3 2 3 = 2 ∙ 3 ∙ 2 1 3 ∙ 3 2 3 = 6�2 ∙ 323 = 6√18 3 Example 2: �𝑥𝑥8 ∙ 𝑦𝑦5 ∙ 𝑧𝑧34 = 𝑥𝑥2 𝑦𝑦�𝑦𝑦𝑧𝑧34 8 4 = 2; 5 4 = 1 + 1 4 To introduce factors into a radical, do the inverse operation. Axel Cotón Gutiérrez Mathematics 4º ESO 4.2.8
- 9. Unit 02 October MATH VOCABULARY: Factor, To Factorize, To Extract, Equal, Improper Faction, Inverse. 4.4. ADDING AND SUBTRACTING RADICALS. Two radicals can only be added (or subtracted) when they have the same index and radicand, that is, when they are similar radicals. 3√5 + 2√5 − √5 = (3 + 2 − 1)√5 = 4√5 2√12 − 3√75 + √27 = 2�22 ∙ 3 − 3�3 ∙ 52 + �33 = 4√3 − 15√3 + 3√3 = −8√3 MATH VOCABULARY: Similar Radicals. 4.5. MULTIPLYING AND DIVIDING RADICALS. Due to the powers properties to multiply radicals with the same index, multiply the radicands and the index remains the same. √5 ∙ √10 = 5 1 2 ∙ ∙ 10 1 2 = (5 ∙ 10) 1 2 = √5 ∙ 10 = √50 = �2 ∙ 52 = 5√2 To multiply radicals with different index, reduce to a common index and then multiply. √𝑎𝑎 ∙ � 𝑎𝑎34 = � 𝑎𝑎48 ∙ � 𝑎𝑎68 = � 𝑎𝑎4 ∙ 𝑎𝑎68 = � 𝑎𝑎108 = 𝑎𝑎� 𝑎𝑎28 = 𝑎𝑎√𝑎𝑎 4 To divide radicals we have to use the same rules, but dividing instead multiplying. Axel Cotón Gutiérrez Mathematics 4º ESO 4.2.9
- 10. Unit 02 October 4.6. POWERS AND ROOTS OF RADICALS. To calculate the power or the root of a radical we express the radicals as powers and we use the power of power property. �√5� 2 = �5 1 2� 2 = 5 1∙2 2 = 5 �√2 34 = �2 1 3 4 = �2 1 3� 1 4 = 2 1 12 = √2 12 5. RATIONALIZING THE DENOMINATOR. Sometimes in Algebra it is desirable to find an equivalent expression for a radical expression that doesn’t have any radicals in the denominator. This process is called rationalizing the denominator. We will use the multiplication identity property. MATH VOCABULARY: To Rationalize, Algebra. 5.1. THE DENOMINATOR IS A SINGLE SQUARE ROOT. When you have a single square root in the denominator you just multiply top and bottom by it. 2 √5 = 2 √5 ∙ √5 √5 = 2√5 5 5.2. THE DENOMINATOR IS A SINGLE NTH ROOT. When you have a single nth root in the denominator, multiply by something so that you will get a perfect nth power. 1 √725 = 1 √725 ∙ √735 √735 = √735 7 Axel Cotón Gutiérrez Mathematics 4º ESO 4.2.10
- 11. Unit 02 October In general: 𝐚𝐚 √𝐛𝐛𝐤𝐤𝐧𝐧 = 𝐚𝐚 √𝐛𝐛𝐤𝐤𝐧𝐧 ∙ √𝐛𝐛𝐧𝐧−𝐤𝐤𝐧𝐧 √𝐛𝐛𝐧𝐧−𝐤𝐤𝐧𝐧 = 𝐚𝐚√𝐛𝐛𝐧𝐧−𝐤𝐤𝐧𝐧 𝐛𝐛 5.3. THE DENOMINATOR IS EITHER A SUM OR A DIFFERENCE OF SQUARE ROOTS. When you have a sum or a difference of square roots (Binomial) in the denominator, multiply the top and the bottom by the conjugate of the denominator. The conjugate of √𝐚𝐚 + √𝐛𝐛 is √𝐚𝐚 − √𝐛𝐛, and vice versa. 1 √2 − √3 = 1 √2 − √3 ∙ √2 + √3 √2 + √3 = √2 + √3 2 − 3 = −√2 − √3 MATH VOCABULARY: Binomial, Conjugate. 6. SCIENTIFIC NOTATION. Scientific Notation (also called Standard Form in Britain) is a special way of writing numbers. It is a number of the form 𝐚𝐚 ∙ 𝟏𝟏𝟏𝟏𝐤𝐤 , where: • |𝐚𝐚| ∈ [ 𝟏𝟏, 𝟏𝟏𝟏𝟏) • 𝒌𝒌 ∈ ℤ, called Order of magnitude. 150,000,000 = 1.5 ∙ 108 0.0000081 = 8.1 ∙ 10−6 It is supposed you know already how to add, subtract, multiply and divide numbers in scientific notation from previous courses MATH VOCABULARY: Scientific Notation, Standard Form, Order of Magnitude. Axel Cotón Gutiérrez Mathematics 4º ESO 4.2.11
- 12. Unit 02 October 7. LOGARITHMS. Given two numbers, 𝒂𝒂 ∈ ℝ+ , 𝒂𝒂 ≠ 𝟏𝟏 and 𝐛𝐛 ∈ ℝ+ , the logarithm in base 𝒂𝒂 of b, 𝐥𝐥𝐨𝐨𝐠𝐠𝐚𝐚 𝐛𝐛, is the index we need for raising 𝒂𝒂 to get 𝒃𝒃. 𝐥𝐥𝐥𝐥𝐥𝐥𝐚𝐚 𝐛𝐛 = 𝐜𝐜 𝐢𝐢𝐢𝐢 𝐚𝐚𝐜𝐜 = 𝐛𝐛 𝑙𝑙𝑙𝑙𝑙𝑙2 8 = 3 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 23 = 8 If the base is 10 (scientific notation), it is called Common Logarithm, and we write it without the base. log100 = 2 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 102 = 100 If the base if the irrational number e, it is called Napierian logarithm. and we write it 𝐥𝐥𝐥𝐥. ln 1 𝑒𝑒3 = ln 𝑒𝑒−3 = −3 MATH VOCABULARY: Logarithm, To Raise, Common Logarithm, Napierian Logarithm. 8. LOGARITHMS PROPERTIES. The logarithms have also some properties that will help us to solve exercises and problems. All of them come from the power properties. 8.1. DIRECT PROPERTIES. 𝐥𝐥𝐥𝐥𝐥𝐥𝐚𝐚 𝟏𝟏 = 𝟎𝟎, 𝐛𝐛𝐛𝐛𝐛𝐛𝐛𝐛𝐛𝐛𝐛𝐛𝐛𝐛𝐛𝐛 𝒂𝒂𝟎𝟎 = 𝟏𝟏 𝐥𝐥𝐥𝐥𝐥𝐥𝐚𝐚 𝐚𝐚 = 𝟏𝟏, 𝐛𝐛𝐛𝐛𝐛𝐛𝐛𝐛𝐛𝐛𝐛𝐛𝐛𝐛𝐛𝐛 𝒂𝒂𝟏𝟏 = 𝒂𝒂 Axel Cotón Gutiérrez Mathematics 4º ESO 4.2.12
- 13. Unit 02 October 8.2. LOGARITHM OF A PRODUCT. The logarithm of a product is the addition of the logarithms of each factor. 𝐥𝐥𝐥𝐥𝐥𝐥𝐚𝐚( 𝐛𝐛 ∙ 𝐜𝐜) = 𝐥𝐥𝐥𝐥𝐥𝐥𝒂𝒂 𝒃𝒃 + 𝐥𝐥𝐥𝐥𝐥𝐥𝒂𝒂 𝒄𝒄 8.3. LOGARITHM OF A FRACTION. The logarithm of a fraction is the subtraction of the logarithms of each factor. 𝐥𝐥𝐥𝐥𝐥𝐥𝐚𝐚 � 𝐛𝐛 𝒄𝒄 � = 𝐥𝐥𝐥𝐥𝐥𝐥𝒂𝒂 𝒃𝒃 − 𝐥𝐥𝐥𝐥𝐥𝐥𝒂𝒂 𝒄𝒄 8.4. LOGARITHM OF A POWER. The logarithm of a power is the product of the index by the logarithm of the power´s base. 𝐥𝐥𝐥𝐥𝐥𝐥𝐚𝐚 𝐛𝐛𝒏𝒏 = 𝐧𝐧 ∙ 𝐥𝐥𝐥𝐥𝐥𝐥𝒂𝒂 𝒃𝒃 8.5. CHANGING BASES. To change the base of a logarithm we use this formula: 𝐥𝐥𝐥𝐥𝐥𝐥𝐚𝐚 𝐛𝐛 = 𝐥𝐥𝐥𝐥𝐥𝐥𝒄𝒄 𝒃𝒃 𝐥𝐥𝐥𝐥𝐥𝐥𝒄𝒄 𝒂𝒂 𝑙𝑙𝑙𝑙𝑙𝑙4 12 = log12 log4 Axel Cotón Gutiérrez Mathematics 4º ESO 4.2.13
- 14. Unit 02 October 8.6. LOGARITHMS EQUATIONS. To solve logarithms equations we have to apply the logarithms and power properties. 𝑙𝑙𝑙𝑙𝑙𝑙9 𝑥𝑥 = 1 4 ⇒ 9 1 4 = 𝑥𝑥 ⇒ (32) 1 4 = 3 2 4 = 3 1 2 = √3 = 𝑥𝑥 Axel Cotón Gutiérrez Mathematics 4º ESO 4.2.14
- 15. Unit 02 October 1. Number each of the boxes below, 1-16. On a separate sheet of paper, show work for each box as you solve the logarithm equation. Each box has at least one equation. Show all your work. 2. On the puzzle below, write your answer for x next to each equation. Then cut out each box individually. 3. To “solve” the puzzle, the touching edges should be equivalent. For example, 2log 4x = should touch the edge where 16x = . 4. Glue down your pieces on a separate sheet of paper. The final product you turn in should be the re-arranged puzzle pieces (should still look like a square) and the page of work you did for all 16 individual squares. Axel Cotón Gutiérrez Mathematics 4º ESO 4.2.15