Lesson 33. Operations with Scientific Notation. Review: Lesson 2A on Base 10, Rules: 33A Converting to and from Scientific Notation

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1 Lesson 33 Operations with Scientific Notation Review: Lesson 2A on Base 10, Rules: When converting to scientific notation, moving the decimal place to the left adds positive values to the base 10 exponent. Moving the decimal to the right adds negative values to the base 10 exponent. When converting from scientific notation, a positive base 10 exponent tells you to move the decimal place to the right. A negative base 10 exponent tells you to move the decimal place to the left. 33A Converting to and from Scientific Notation Many people, scientists especially, often have to work with extremely large, or extremely small quantities. In chemistry, a common unit is the mole. Like one dozen refers to a quantity of 12, 1 mole refers to a quantity of 602,21,129,000,000,000,000,000. That is a lot of zeros to keep up with! Imagine if you had to write the entire number every time you needed it? The chances of making an error would be great. That s where scientific notation comes in handy. It allows us to write really large and really small numbers in a more efficient way. Do you remember writing numbers in expanded form when you were studying arithmetic? For example, the number 5,378 in expanded form equals 5 x x x x Scientific notation is similar. Taking 5,378 as an example again, first we would write it with a decimal place as Then, we would move the decimal place to the left 3 places, until it was to the right of the last non-zero digit, which is 5. In scientific notation, it would look like this: x 10 3 We could convert it back to standard form by moving the decimal to the right 3 places. This works because 10 3 equals 1,000, so all we are doing is multiplying by 1,000. Now, what if we had x 10-3? The base 10 exponent is a -3 instead of a +3. How would we write that in standard form? Well, just like we moved

2 the decimal to the right 3 when the exponent was +3, we will move the decimal the left three since it s a -3, and get: x 10-3 = To convert back to scientific notation, we just move the decimal to the right of the first non-zero digit. Notice that moving to the right causes us to add negative values to the base 10 exponent. Example 33.1 Convert the following to scientific notation, and round to 2 decimal places: a),285 b) c) d) a) If no decimal point is shown, place one where it should go, then move it accordingly:,285 = 285. =.285 = moved 3 places left so =.285 x 10 3 =.29 x 10 3 b) = = moved places left so = x 10 = 6.22 x 10 c) = 5.91 = moved places right so = 5.91 x 10 - d) = 6.09 = moved 3 places right so = 6.09 x 10-3 Example 33.2 Convert the following from scientific notation to standard form: a) 3.11 x 10 b) 21 x 10 5 c) 7.08 x 10-2 d) 65. x 10-5 a) the + indicates the decimal place needs to be moved places to the right: 3.11 = = 31,100 b) Standard sci. notation form has a decimal point to the right of the first nonzero digit, so this problem is not in standard form. Don t worry about that though, just put a decimal point where it would normally go, and move 5 places to the right: 21. = = 2,100,000 You can check your work by moving the decimal back to the left 5 places, which, if you did it correctly, should put it to the right of the 1. c) The -2 indicates the decimal place needs to move 2 places to the left: 7.08 = 0.078

3 d) Like (c), this is not in standard form, but don t worry about that. Just start at the decimal point and move to the left 5 places: 65. = B Operations with Scientific Notation Just use the same rules for working with exponents that you ve been using in Lessons 30-32, and apply them now to scientific notation problems. Example 33.3 Multiply. a) ( x 10 ) (3 x 10-2 ) b) (6 x 10 8 )(3 x 10 7 ) Recognize that you have factors in each of these problems. First, rearrange them so the base 10 factors are together, then add the exponents. Multiply the remaining 2 factors, and then write the answer in standard form (decimal to the right of the first non-zero digit). a) ()(3)(10 )(10-2 ) = 12 x 10-2 = 12 x 10 2 = 1.2 x 10 3 b) (6)(3)(10 8 )(10 7 ) = 18 x = 18 x x Example 33. Divide. a) b) Simplify by dividing non-base 10 numbers, while using the quotient rule for exponents (Lesson 7) on the base 10 numbers. a) Simplify 8 2 =, and apply the quotient rule for exponents to get: ()(10 )(10 2 ) = ()(10 +2 ) = x 10 6 b) Simplify 6 3 = 2, and apply the quotient rule for exponents to get: (2)(10 8 )(10-11 ) = (2)( ) = 2 x 10-3

4 Practice Set 33 Use your best judgement as to when you should and shouldn t use a calculator Convert the following to scientific notation, and round to 2 decimal places: Convert the following to scientific notation, and round to 2 decimal places: Convert the following from scientific notation to standard form:.851 x Convert the following from scientific notation to standard form: 6.19 x Multiply: (3 x 10-1 )(5 x 10 5 ) Divide: x 5 x 2x 5 2x 3 t 2 t + 6s 6 s 3 t 3 2t i i 3 i Which of the following, when simplified, results in a negative number? A) (-3) 3 B) (-3) 2 C) A and B D) Neither A or B 12 29,. (CLEP College Algebra) Which of the following numbers are irrational? I. 0.3 II. π III. A) I only B) II only C) I and II only D) II and III only E) I, II, and III The 1912 eruption of Novarupta volcano in Alaska was 30x larger than Mt. St. Helens 1980 eruption. However, an Ice Age eruption in the Yellowstone Caldera was 2,00 times larger than Mt. St. Helens! How many times larger was the Yellowstone Ice Age eruption than the 1912 Novarupta eruption? Find 3n +1 n=1

5 15 2. What is the probability of the number spinner landing on the? For the function y=x 2, estimate the slope of the line tangent to the function at x =1. In other words, find f (x) at x=1. Arrive at the slope by using the increasingly smaller values of x shown in the table. The first 3 rows of the table have been done for you. x y =f(x+ x) - f(x) Δy Δx 1 f(2) - f(1) = = -1=3 3/1 = f(1.5) - f(1) = = = /0.5 = f(1.1) - f(1) = = = /0.1 = f(1.01) - f(1) = = = /0.01 =? Find x(-1) when x(t) = (t+2)(t-3) Rearrange the steps shown below in the correct order to complete the proof of Euclid s Proposition 1. Don t rewrite everything, just write the letters A,B,C, etc. in the correct order. A)With center A and distance AB, construct a circle BCD[Post. 3]. B) With center B and distance BA, construct a circle ACE[Post. 3]. C) Construct line segment AB[Given]. D) Now, since point A is the center of circle CBD, AC = AB [Def. of circle]. E) Also, since point B is the center of circle ACE, BC = BA [Def. of circle]. F) Therefore, the three line segments CA, AB, and BC are equal to each other, and triangle ABC is therefore an equilateral triangle, being what was required to do. G) From point C, where the circles intersect, construct line segments CA and CB[Post. 1]. H) But CA was also proved equal to AB. Therefore, CA must also equal CB, since things equal to the same thing are equal to each other [Axiom 1] Turn the following sentence into an equation and solve it Simplify 36 The product of 7 and the opposite of a number is -21.