Properties of Exponents

Transcription

1 Slide 1 / 234 Slide 2 / 234 Properties of Exponents Return to Table of ontents Slide 3 / 234 Properties of Exponents Examples Slide 4 / 234 Slide 5 / 234 Slide 6 / Simplify the expression: 2 Simplify the expression:

2 Slide 7 / 234 Slide 8 / Simplify the expression: Slide 9 / 234 Slide 10 / Simplify the expression: Slide 11 / 234 Slide 12 / 234 Writing Numbers in Scientific Notation Writing Large Numbers in Scientific Notation

3 Slide 13 / 234 Here are some different ways of writing 6,500. 6,500 = 6.5 thousand 6.5 thousand = 6.5 x 1, x 1,000 = 6.5 x 10 3 which means that 6,500 = 6.5 x 10 3 Scientific Notation 6,500 is the standard form of the number 6.5 x 10 3 is scientific notation Slide 14 / 234 Scientific Notation 6.5 x 10 3 isn't a lot more convenient than 6,500. ut let's do the same thing with 7,400,000,000 which is equal to 7.4 billion which is 7.4 x 1,000,000,000 which is 7.4 x 10 9 esides being shorter than 7,400,000,000, its a lot easier to keep track of the zeros in scientific notation. nd we'll see that the math gets a lot easier as well. These are two ways of writing the same number. Slide 15 / 234 Scientific Notation Slide 16 / 234 Express 870,000 in scientific notation Scientific notation expresses numbers as the product of: a coefficient and 10 raised to some power x 10 6 The coefficient is always greater than or equal to one, and less than 10. In this case, the number 3,780,000 is expressed in scientific notation. 1. Write the number without the comma. 2. Place the decimal so that the first number will be less than 10 but greater than or equal to ount how many places you had to move the decimal point. This becomes the exponent of rop the zeros to the right of the rightmost non-zero digit x x x 10 5 Slide 17 / 234 Express 53,600 in scientific notation 1. Write the number without the comma. 2. Place the decimal so that the first number will be less than 10 but greater than or equal to ount how many places you had to move the decimal point. This becomes the exponent of rop the zeros to the right of the rightmost non-zero digit x x x 10 4 Slide 18 / 234 Express 284,000,000 in scientific notation 1. Write the number without the comma. 2. Place the decimal so that the first number will be less than 10 but greater than or equal to ount how many places you had to move the decimal point. This becomes the exponent of rop the zeros to the right of the rightmost non-zero digit x x x

4 Slide 19 / Which is the correct coefficient of 147,000 when it is written in scientific notation Slide 20 / Which is the correct coefficient of 23,400,000 when it is written in scientific notation Slide 21 / How many places do you need to move the decimal point to change 190,000 to 1.9 Slide 22 / How many places do you need to move the decimal point to change 765,200,000,000 to Slide 23 / Which of the following is 345,000,000 in scientific notation 3.45 x x x x 10 9 Slide 24 / Which of these is not a large number in scientific notation.34 x x x x 10-1 E 11.4 x F.41 x 10 3 G 5.65 x 10 4 H 10.0 x 10 3

5 Slide 25 / 234 The ontent/writing mass of the solar space system is 300,000,000,000,000,000,000,000,000,000,000,000,000,000, 000,000,000,000 kg (How do you say that number) Slide 26 / 234 (That's 3, followed by 53 zeros) What is this number in scientific notation More Practice Slide 27 / 234 Express 9,040,000,000 in scientific notation Slide 28 / 234 Express 13,030,000 in scientific notation 1. Write the number without the comma. 2. Place the decimal so that the first number will be less than 10 but greater than or equal to x Write the number without the comma. 2. Place the decimal so that the first number will be less than 10 but greater than or equal to x ount how many places you had to move the decimal point. This becomes the exponent of x ount how many places you had to move the decimal point. This becomes the exponent of x rop the zeros to the right of the rightmost non-zero digit x rop the zeros to the right of the rightmost non-zero digit x 10 7 Slide 29 / 234 Express 1,000,000,000 in scientific notation 1. Write the number without the comma. 2. Place the decimal so that the first number will be less than 10 but greater than or equal to ount how many places you had to move the decimal point. This becomes the exponent of x x Slide 30 / Which of the following is 12,300,000 in scientific notation.123 x x x x rop the zeros to the right of the rightmost non-zero digit. 1 x 10 9

6 Slide 31 / 234 Slide 32 / 234 Express in scientific notation Writing Small Numbers in Scientific Notation 1. Write the number without the decimal point. 2. Place the decimal so that the first number is 1 or more, but less than ount how many places you had to move the decimal point. The negative of this numbers becomes the exponent of rop the zeros to the right of the rightmost non-zero digit x x x 10-3 Slide 33 / 234 Slide 34 / 234 Express in scientific notation Express in scientific notation 1. Write the number without the comma Write the number without the comma Place the decimal so that the first number will be less than 10 but greater than or equal to x Place the decimal so that the first number will be less than 10 but greater than or equal to x ount how many places you had to move the decimal point. This becomes the exponent of x ount how many places you had to move the decimal point. This becomes the exponent of x rop the zeros to the right of the rightmost non-zero digit x rop the zeros to the right of the rightmost non-zero digit. 7.3 x 10-3 Slide 35 / 234 Slide 36 / Which is the correct decimal placement to convert to scientific notation Which is the correct decimal placement to convert to scientific notation

7 Slide 37 / How many times do you need to move the decimal point to change to 6.58 Slide 38 / How many times do you need to move the decimal point to change to Slide 39 / Write in scientific notation x x x x 10-3 Slide 40 / Which of these is not a small number in scientific notation.34 x x x x 10-1 E 11.4 x F.41 x 10-3 G 5.65 x 10-4 H 10.0 x 10-3 Slide 41 / 234 Slide 42 / 234 Express in scientific notation More Practice 1. Write the number without the comma. 2. Place the decimal so that the first number will be less than 10 but greater than or equal to ount how many places you had to move the decimal point. This becomes the exponent of rop the zeros to the right of the right-most non-zero digit x x x 10-3

8 Slide 43 / 234 Express in scientific notation Slide 44 / 234 Express in scientific notation 1. Write the number without the comma Write the number without the comma Place the decimal so that the first number will be less than 10 but greater than or equal to x Place the decimal so that the first number will be less than 10 but greater than or equal to x ount how many places you had to move the decimal point. This becomes the exponent of x ount how many places you had to move the decimal point. This becomes the exponent of x rop the zeros to the right of the right-most non-zero digit. 9.2 x rop the zeros to the right of the right-most non-zero digit. 1.2 x 10-6 Slide 45 / 234 Slide 46 / Write in scientific notation x x x x 10-5 onverting to Standard Form Slide 47 / 234 Slide 48 / 234 Express 3.5 x 10 4 in standard form Express 1.02 x 10 6 in standard form 1. Write the coefficient Write the coefficient dd a number of zeros equal to the exponent: to the right for positive exponents and to the left for negative dd a number of zeros equal to the exponent: to the right for positive exponents and to the left for negative Move the decimal the number of places indicated by the exponent: to the right for positive exponents and to the left for negative Move the decimal the number of places indicated by the exponent: to the right for positive exponents and to the left for negative rop unnecessary zeros and add comma, as necessary. 35, rop unnecessary zeros and add comma, as necessary. 1,020,000

9 Slide 49 / 234 Express 3.45 x 10-3 in standard form Slide 50 / 234 Express 2.95 x 10-5 in standard form 1. Write the coefficient Write the coefficient dd a number of zeros equal to the exponent: to the right for positive exponents and to the left for negative dd a number of zeros equal to the exponent: to the right for positive exponents and to the left for negative Move the decimal the number of places indicated by the exponent: to the right for positive exponents and to the left for negative Move the decimal the number of places indicated by the exponent: to the right for positive exponents and to the left for negative rop unnecessary zeros and add comma, as necessary. 4. rop unnecessary zeros and add comma, as necessary. Slide 51 / 234 Slide 52 / How many times do you need to move the decimal and which direction to change 7.41 x 10-6 into standard form 22 How many times do you need to move the decimal and which direction to change 4.5 x into standard form 6 to the right 6 to the left 7 to the right 7 to the left 10 to the right 10 to the left 11 to the right 11 to the left Slide 53 / Write 6.46 x 10 4 in standard form. Slide 54 / Write 3.4 x 10 3 in standard form. 646, , , ,

10 Slide 55 / Write 6.46 x 10-5 in standard form. Slide 56 / Write 1.25 x 10-4 in standard form. 646, Slide 57 / Write 4.56 x 10-2 in standard form Slide 58 / Write 1.01 x 10 9 in standard form. 101,000,000,000 1,010,000, Slide 59 / 234 Slide 60 / 234 omparing numbers in scientific notation omparing Numbers Written in Scientific Notation First, compare the exponents. If the exponents are different, the coefficients don't matter; they have a smaller effect. Whichever number has the larger exponent is the larger number.

11 Slide 61 / 234 omparing numbers in scientific notation When the exponents are the different, just compare the exponents. < = > 9.99 x x 10 4 Slide 62 / 234 omparing numbers in scientific notation If the exponents are the same, compare the coefficients. The larger the coefficient, the larger the number (if the exponents are the same) x x x x 10-2 Slide 63 / 234 omparing numbers in scientific notation Slide 64 / Which is ordered from least to greatest When the exponents are the same, just compare the coefficients. < = > I, II, III, IV IV, III, I, II I, IV, II, III I. 1.0 x 10 5 II. 7.5 x 10 6 III. 8.3 x x x 10 3 III, I, II, IV IV. 5.4 x x x x x Slide 65 / 234 Slide 66 / Which is ordered from least to greatest 31 Which is ordered from least to greatest I, II, III, IV I. 1.0 x 10 2 I, II, III, IV I. 1 x 10 2 IV, III, I, II II. 7.5 x 10 6 IV, III, I, II II. 7.5 x 10 3 I, IV, II, III I, II, IV, III III. 8.3 x 10 9 IV. 5.4 x 10 7 III, IV, II, I III, IV, I, II III. 8.3 x 10-2 IV. 5.4 x 10-3

12 Slide 67 / Which is ordered from least to greatest Slide 68 / Which is ordered from least to greatest II, III, I, IV I. 1 x 10-2 I, II, III, IV I. 1.0 x 10 2 IV, III, I, II II. 7.5 x IV, III, I, II II. 7.5 x 10 2 III, IV, II, I III. 8.3 x I, IV, II, III III. 8.3 x 10 2 III, IV, I, II IV. 5.4 x 10 2 III, IV, I, II IV. 5.4 x 10 2 Slide 69 / 234 Slide 70 / Which is ordered from least to greatest 35 Which is ordered from least to greatest I, II, III, IV I. 1.0 x 10 6 I, II, III, IV I. 1.0 x 10 3 IV, III, I, II II. 7.5 x 10 6 IV, III, I, II II. 5.0 x 10 3 I, IV, II, III III. 8.3 x 10 6 I, IV, II, III III. 8.3 x 10 6 III, IV, I, II IV. 5.4 x 10 7 III, IV, I, II IV. 9.5 x 10 6 Slide 71 / Which is ordered from least to greatest I, II, III, IV I. 2.5 x 10-3 Slide 72 / 234 Multiplying Numbers in Scientific Notation IV, III, I, II I, IV, II, III III, IV, I, II II. 5.0 x 10-3 III. 9.2 x 10-6 IV. 4.2 x 10-6 Multiplying with scientific notation requires at least three, and sometimes four, steps. 1. Multiply the coefficients 2. Multiply the powers of ten 3. ombine those results 4. Put in proper form

13 Slide 73 / 234 Multiplying Numbers in Scientific Notation Evaluate: (6.0 x 10 4 )(2.5 x 10 2 ) Slide 74 / 234 Multiplying Numbers in Scientific Notation Evaluate: (4.80 x 10 6 )(9.0 x 10-8 ) 1. Multiply the coefficients 6.0 x 2.5 = Multiply the coefficients 4.8 x 9.0 = Multiply the powers of ten 10 4 x 10 2 = Multiply the powers of ten 10 6 x 10-8 = ombine those results 15 x ombine those results 43.2 x Put in proper form 1.5 x Put in proper form 4.32 x 10-1 Slide 75 / Evaluate (2.0 x 10-4 )(4.0 x 10 7 ). Express the result in scientific notation. 8.0 x x x x E 7.68 x F 7.68 x Slide 76 / Evaluate (5.0 x 10 6 )(7.0 x 10 7 ) 3.5 x x x x 10-1 E 7.1 x F 7.1 x Evaluate (6.0 x 10 2 )(2.0 x 10 3 ) 1.2 x x x x 10-1 E 3.0 x 10 5 F 3.0 x 10 1 Slide 77 / 234 Slide 78 / Evaluate (1.2 x 10-6 )(2.5 x 10 3 ). Express the result in scientific notation. 3 x x x x E 30 x 10 18

14 Slide 79 / 234 Slide 80 / Evaluate (1.1 x 10 4 )(3.4 x 10 6 ). Express the result in scientific notation x x x x E 37.4 x Evaluate (3.3 x 10 4 )(9.6 x 10 3 ). Express the result in scientific notation x x x x 10 8 E 30 x 10 7 Slide 81 / Evaluate (2.2 x 10-5 )(4.6 x 10-4 ). Express the result in scientific notation. Slide 82 / 234 ividing Numbers in Scientific Notation x x x x 10-9 E x 10-8 ividing with scientific notation is just like multiplying. 1. ivide the coefficients 2. ivide the powers of ten 3. ombine those results 4. Put in proper form Slide 83 / 234 ivision with Scientific Notation Evaluate: 5.4 x x 10 2 Slide 84 / 234 ivision with Scientific Notation Evaluate: 4.4 x x Multiply the coefficients = Multiply the coefficients = Multiply the powers of ten = Multiply the powers of ten = ombine those results 0.6 x ombine those results 4.0 x Put in proper form 6.0 x Put in proper form

15 Slide 85 / Evaluate 4.16 x x 10-5 Express the result in scientific notation. Slide 86 / Evaluate 7.6 x x 10-4 Express the result in scientific notation. 0.8 x x x x 10-4 E 8 x x x x x 10-8 E 1.9 x 10 8 Slide 87 / Evaluate 8.2 x x 10 7 Express the result in scientific notation. Slide 88 / Evaluate 3.2 x x 10-4 Express the result in scientific notation. 4.1 x x x x E 4.1 x x x x x 10 1 E 5 x 10 3 Slide 89 / The point on a pin has a diameter of approximately 1 x 10-4 meters. If an atom has a diameter of 2 x meters, about how many atoms could fit across the diameter of the point of a pin Slide 90 / 234 ddition and Subtraction with Scientific Notation 50, ,000 2,000,000 5,000,000 Question from P lgebra I End-of-ourse Practice Test Numbers in scientific notation can only be added or subtracted if they have the same exponents. If needed, an intermediary step is to rewrite one of the numbers so it has the same exponent as the other.

16 Slide 91 / 234 ddition and Subtraction Slide 92 / 234 ddition and Subtraction This is the simplest example of addition 4.0 x x 10 3 = Since the exponents are the same (3), just add the coefficients. 4.0 x x 10 3 = 9.3 x 10 3 This just says that if you add 4.0 thousand and 5.3 thousand to get 9.3 thousand. This problem is slightly more difficult because you need to add one extra step at the end. 8.0 x x 10 3 = Since the exponents are the same (3), just add the coefficients. 4.0 x x 10 3 = 13.3 x 10 3 ut that is not proper form, since 13.3 > 10; it should be written as 1.33 x x x 10 3 = Slide 93 / 234 ddition and Subtraction This requires an extra step at the beginning because the exponents are different. We have to either convert the first number to 80 x 10 3 or the second one to 0.53 x The latter approach saves us a step at the end. 8.0 x x 10 4 = 8.53 x 10 4 Slide 94 / The sum of 5.6 x 10 3 and 2.4 x 10 3 is 8.0 x x x x 10 3 Once both numbers had the same exponents, we just add the coefficient. Note that when we made the exponent 1 bigger, that's makes the number 10x bigger; we had to make the coefficient 1/10 as large to keep the number the same. Slide 95 / x 10 3 minus 2.0 x 10 3 is 6.0 x x x x 10 3 Slide 96 / x 10 3 plus 2.0 x 10 2 is 9.0 x x x x 10 2

17 Slide 97 / 234 Slide 98 / x 10 5 plus 7.8 x 10 5 is 11.3 x x x x Slide 99 / 234 Slide 100 / 234 The symbol for taking a square root is, it is a radical sign. The square root cancels out the square. There is no real square root of a negative number. Roots is not real (4 2 =16 and (-4) 2 =16) Slide 101 / 234 Slide 102 / What is 1 54 What is

18 Slide 103 / 234 Slide 104 / What is Slide 105 / 234 Slide 106 / 234 To take the square root of a variable rewrite its exponent as the square of a power. Square roots need to be positive answers. Even powered answered, like above, are positive even if the variables negative. The same cannot be said if the answer has an odd power. When you take a square root an the answer has an odd power, put the answer inside of absolute value signs. Slide 107 / 234 Slide 108 / 234

19 Slide 109 / 234 Slide 110 / 234 Slide 111 / 234 Slide 112 / no real solution Slide 113 / 234 Slide 114 / no real solution no real solution

20 Slide 115 / 234 Slide 116 / no real solution Slide 117 / 234 Slide 118 / Evaluate 67 Evaluate No Real Solution No Real Solution Slide 119 / 234 Slide 120 / Evaluate No Real Solution

21 Slide 121 / 234 Q: If a square root cancels a square, what cancels a cube : cube root. Slide 122 / 234 The volume (V) of a cube is found by cubing its side length (s). V = s 3 V = s 3 V = 4 3 = V = 64 cubic units or 64 units 3 4 units The volume (V) of a cube is labeled as cubic units, or units 3, because to find the volume, you need to cube its side. Slide 123 / 234 Slide 124 / 234 cube with sides 3 units would have a volume of 27 u 3 because 3 3 =27. If a cube has an volume of 64 u 3 what is the length of one side Need to find a number when multiplied by itself three times will equal = 64, so 4 units is the length of a side. Slide 125 / 234 Slide 126 / 234

22 Slide 127 / 234 Slide 128 / 234 Slide 129 / 234 Slide 130 / 234 Slide 131 / 234 Slide 132 / 234

23 Slide 133 / 234 Slide 134 / Simplify not possible Slide 135 / 234 Slide 136 / Simplify not possible Slide 137 / 234 Slide 138 / Which of the following is not a step in simplifying

24 Slide 139 / 234 Slide 140 / 234 In general, and absolute value signs are needed if n is even and the variable has an odd powered answer. Slide 141 / 234 Slide 142 / Simplify Slide 143 / 234 Slide 144 / Simplify

25 Slide 145 / 234 Slide 146 / Simplify 86 Simplify Slide 147 / 234 Slide 148 / Simplify 88 If the n th root of a radicand is, which of the following is always true No absolute value signs are ever needed. bsolute value signs will always be needed. bsolute value signs will be needed if j is negative. bsolute value signs are needed if n is an even index. Slide 149 / 234 Slide 150 / 234 Simplifying Radicals is said to be a rational answer because their is a perfect square that equals the radicand. If a radicand doesn't have a perfect square that equals it, the root is said to be irrational. The square root of the following numbers is rational or irrational

26 Slide 151 / 234 Slide 152 / 234 The commonly excepted form of a radical is called the "simplified form To simplify a non-perfect square, start by breaking the radicand into factors and then breaking the factors into factors and so on until there only prime numbers are left. this is called the prime factorization. Slide 153 / Which of the following is the prime factorization of 24 3(8) 4(6) 2(2)(2)(3) 2(2)(2)(3)(3) Slide 154 / Which of the following is the prime factorization of 72 9(8) 2(2)(2)(2)(6) 2(2)(2)(3) 2(2)(2)(3)(3) Slide 155 / Which of the following is the prime factorization of 12 3(4) 2(6) 2(2)(2)(3) 2(2)(3) Slide 156 / Which of the following is the prime factorization of 24 rewritten as powers of factors

27 Slide 157 / 234 Slide 158 / Which of the following is the prime factorization of 72 rewritten as powers of factors Slide 159 / 234 Slide 160 / Simplify already in simplified form Slide 161 / 234 Slide 162 / Simplify 96 Simplify already in simplified form already in simplified form

28 Slide 163 / 234 Slide 164 / Simplify 98 Which of the following does not have an irrational simplified form already in simplified form Slide 165 / 234 Simplifying Roots of Variables ivide the index into the exponent. The number of times the index goes into the exponent becomes the power on the outside of the radical and the remainder is the power of the radicand. Slide 166 / 234 Simplifying Roots of Variables What about the absolute value signs n bsolute Value sign is needed if the index is even, the starting power of the variable is even and the answer is an odd power on the outside. Examples of when absolute values are needed: Slide 167 / 234 Slide 168 / 234

29 Slide 169 / 234 Slide 170 / Simplify Slide 171 / 234 Slide 172 / Simplify Operations with Radicals To add and subtract radicals they must be like terms. Radicals are like terms if they have the same radicands and the same indexes. Like Terms Unlike Terms Slide 173 / Identify all of the pairs of like terms Slide 174 / 234 To add or subtract radicals, only the coefficients of the like terms are combined. E F

30 Slide 175 / 234 Slide 176 / Simplify lready Simplified Slide 177 / 234 Slide 178 / Simplify lready Simplified Slide 179 / 234 Slide 180 / Simplify Some irrational radicals will not be like terms, but can be simplified. In theses cases, simplify then check for like terms. lready Simplified

31 Slide 181 / 234 Slide 182 / Simplify lready in simplest form Slide 183 / 234 Slide 184 / Simplify lready in simplest form Slide 185 / 234 Slide 186 / Which of the following expressions does not equal the other 3 expressions

32 Slide 187 / 234 Slide 188 / Multiply Multiplying Square Roots fter multiplying, check to see if radicand can be simplified. Slide 189 / 234 Slide 190 / Simplify 116 Simplify Slide 191 / 234 Slide 192 / Simplify 118 Simplify

33 Slide 193 / 234 Multiplying Polynomials Involving Radicals 1) Follow the rules for distribution. 2)e sure to simplify radicals when possible and combine like terms. Slide 194 / Multiply and write in simplest form: Slide 195 / Multiply and write in simplest form: Slide 196 / Multiply and write in simplest form: Slide 197 / Multiply and write in simplest form: Slide 198 / Multiply and write in simplest form:

34 Slide 199 / 234 Rationalizing the enominator Mathematicians don't like radicals in the denominators of fractions. When there is one, the denominator is said to be irrational. The method used to rid the denominator is termed "rationalizing the denominator". Which of these has a rational denominator Slide 200 / 234 If a denominator needs to be rationalized, start by finding its conjugate. conjugate is another polynomial that when the conjugate and the denominator are multiplied, no more irrational term. The conjugate for a monomial with a square root is the same square root. Example has a conjugate of. Why ecause Rational enominator Irrational enominator The conjugate of a binomial with square roots is the opposite operation between the terms. Example has a conjugate of. Why ecause Slide 201 / 234 an you find a pattern for when a binomial is multiplied by its conjugate Example Example Example Slide 202 / What is conjugate of o you see a pattern that let's us go from line 1 to line 3 directly (term 1) 2 - (term 2) 2 Slide 203 / 234 Slide 204 / What is conjugate of

35 Slide 205 / 234 Slide 206 / What is conjugate of Slide 207 / 234 The goal is to rationalize the denominator without changing the value of the fraction. To do this multiply the numerator and denominator by the same exact value. Rationalize the enominator: Slide 208 / 234 Examples: Why The no original absolute x in the value radicand had an odd signs power. Rationalize the enominator: Slide 209 / Simplify Slide 210 / 234 lready simplified

36 Slide 211 / 234 Slide 212 / Simplify 131 Simplify lready simplified lready simplified Slide 213 / 234 Slide 214 / 234 Slide 215 / 234 Slide 216 / 234 Rationalizing n th roots of monomials Remember that, given an n th root in the denominator, you will need to find the conjugate that makes the radicand to the n th power. Examples: 134 Rationalize

37 Slide 217 / 234 Slide 218 / Rationalize Slide 219 / 234 Slide 220 / 234 Rational Exponents Slide 221 / 234 Rational Exponents, or exponents that are fractions, is another way to write a radical. Slide 222 / 234 Rewrite each radical as a rational exponent in the lowest terms.

38 Slide 223 / 234 Slide 224 / 234 Slide 225 / 234 Slide 226 / Find the simplified expression that is equivalent to: Slide 227 / Find the simplified expression that is equivalent to: Slide 228 / Find the simplified expression that is equivalent to:

39 Slide 229 / 234 Slide 230 / Find the simplified expression that is equivalent to: Slide 231 / 234 Slide 232 / Simplify 145 Simplify Slide 233 / 234 Slide 234 / Simplify