Scientific Notation with a Calculator Andrew Gloag Melissa Kramer Anne Gloag
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AUTHORS Andrew Gloag Melissa Kramer Anne Gloag
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C HAPTER
Chapter 1. Scientific Notation with a Calculator
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Scientific Notation with a Calculator
Here you’ll learn how to use a calculator to simplify expressions in scientific notation. Did you know that there are about 7 billion people living on the Earth? Also, did you know that each person is made up of about 100 trillion cells? To find the total number of cells making up all the people on Earth, you would have to multiply these two numbers together, but without a calculator, that might be difficult. In this Concept, you’ll learn how to evaluate expressions using scientific notation with a calculator so that you can perform multiplication problems such as this one. Guidance
Scientific and graphing calculators make scientific notation easier. To compute scientific notation, use the [EE] button. This is [2nd] [,] on some TI models or [10χ ], which is [2nd] [log]. Example A
Evaluate 2.6 × 105 using a scientific calculator. Solution: To enter 2.6 × 105 , enter 2.6 [EE] 5.
When you hit [ENTER] the calculator displays 2.6E5 if it’s set in Scientific mode OR it displays 260,000 if it’s set in Normal mode. Evaluating Expressions Using Scientific Notation We can use scientific notations in expressions, equations and operations. Scientific notation makes it easier to multiply two large numbers together, or divide a decimal by another decimal. This is because when evaluating expressions with scientific notation, you can keep the powers of 10 together and deal with them separately. Example B
Simplify (3.2 × 106 ) · (8.7 × 1011 ). Solution: Start by evaluating by hand, and then check on your calculator: (3.2 × 106 ) · (8.7 × 1011 ) = 3.2 × 8.7 · 106 × 1011 = 27.84 × 1017 = 2.784 × 101 × 1017 = 2.784 × 1018 1
www.ck12.org It is necessary to keep one number before the decimal point, to be in correct scientific notation. In order to do that, we had to use the fact that 27.84 = 2.784 × 101 so we could simplify the expression fully.
In this case, the power is so large that the answer shows up in scientific notation. It does not matter if the calculator is set to show numbers in scientific notation or normally.
Example C
The mass of a single lithium atom is approximately one percent of one millionth of one billionth of one billionth of one kilogram. Express this mass in scientific notation.
Solution: We know that percent means we divide by 100, and so our calculation for the mass (in kg) is: 1 1 −2 × 10−6 × 10−9 × 10−9 1,000,000,000 × 1,000,000,000 = 10 Next, we use the product of powers rule we learned in a previous Concept.
10−2 × 10−6 × 10−9 × 10−9 = 10((−2)+(−6)+(−9)+(−9)) = 10−26 kg. The mass of one lithium atom is approximately 1 × 10−26 kg. Check the answer on your calculator: 2
1 100
1 × 1,000,000 ×
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Chapter 1. Scientific Notation with a Calculator
Video Review
MEDIA Click image to the left or use the URL below. URL: http://www.ck12.org/flx/render/embeddedobject/79511
MEDIA Click image to the left or use the URL below. URL: http://www.ck12.org/flx/render/embeddedobject/79512
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Guided Practice
Evaluate the following expressions by hand and check your answer using a scientific calculator. (a) (1.7 × 106 ) · (2.7 × 10−11 ) (b) (1.03 × 108 ) ÷ (−3.25 × 10−5 ) Solution: (a) (1.7 × 106 ) · (2.7 × 10−11 ) = 1.7 × 2.7 · 106 × 10−11 = 4.59 × 10−5
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www.ck12.org (b) 1.03 1.03 × 108 108 = × −3.25 × 10−5 −3.25 10−5 8−(−5) = −0.317 × 10 = −0.317 × 1013 = −3.17 × 1012
(1.03 × 108 ) ÷ (−3.25 × 10−5 ) =
You must remember to keep the powers of ten together, and have 1 number before the decimal. Explore More
Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK-12 Ba sic Algebra: Scientific Notation (14:26)
MEDIA Click image to the left or use the URL below. URL: http://www.ck12.org/flx/render/embeddedobject/110
Evaluate the following expressions by hand and check your answers using a graphing calculator. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
(3.2 × 106 ) · (8.7 × 1011 ) (5.2 × 10−4 ) · (3.8 × 10−19 ) (1.7 × 106 ) · (2.7 × 10−11 ) (3.2 × 106 ) ÷ (8.7 × 1011 ) (5.2 × 10−4 ) ÷ (3.8 × 10−19 ) (1.7 × 106 ) ÷ (2.7 × 10−11 ) (4.7 × 103 ) · (2.35 × 10−27 ) (1.05 × 10−16 ) · (7.003 × 1021 ) (2.09 × 105 ) ÷ (5.006 × 10−3 ) (9.01 × 1022 ) ÷ (2.6 × 1033 )
Mixed Review 11. 14 milliliters of a 40% sugar solution was mixed with 4 milliliters of pure water. What is the concentration of the mixture? ( 6x + 3y + 18 12. Solve the system . −15 = 11y − 5x 4
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Chapter 1. Scientific Notation with a Calculator
13. Graph the function by creating a table: f (x) = 2x2 . Use the following values for x : −5 ≤ x ≤ 5. 6 2 −6 14. Simplify 5aab11 bc . Your answer should have only positive exponents. 15. Each year Americans produce about 230 million tons of trash (Source: http://www.learner.org/interactives/ garbage/solidwaste.html ). There are 307,006,550 people in the United States. How much trash is produced per person per year? 16. The volume of a 3-dimesional box is given by the formula: V = l(w)(h), where l = length,w = width, and h = height of the box. The box holds 312 cubic inches and has a length of 12 inches and a width of 8 inches. How tall is the box? Answers for Explore More Problems
To view the Explore More answers, open this PDF file and look for section 8.6.
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