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Section 2.8 Mathematical Operations in Scientific Notation Student Learning Focus • Be able to carry out basic mathematical operations using numbers expressed in scientific notation, without changing them to decimal notation. 1 Section 2.8 Mathematical Operations in Scientific Notation Multiplication and Division in Scientific Notation • • Multiplying and Dividing in scientific notation is pretty straightforward. Fewest number of significant figures in the coefficient. 2 Section 2.8 Mathematical Operations in Scientific Notation Multiplication and Division in Scientific Notation • • • • • (2.05 x 10–3) x (1.19 x 10–7) Calculator gives 2.4395 x 10–10 But this answer has 5 sig figs in the coefficient Your calculator does not know about sig figs The actual answer is 2.44 x 10–10 3 Section 2.7 Expressing Numbers in Scientific Notation Learning Check Solve the following and express the answer in scientific notation to the correct number of significant figures. 1. (2.340 x 10–3) x (2.60 x 106) 1. (8.77 x 10–6) x (7.00 x 10–2) 4 Section 2.7 Expressing Numbers in Scientific Notation Solution Solve the following and express the answer in scientific notation to the correct number of significant figures. 1. (2.340 x 10–3) x (2.60 x 106) 6084 = 6.08x103 2. (8.77 x 10–6) x (7.00 x 10–2) 0.0000006.14 = 6.14x10–7 5 Section 2.8 Mathematical Operations in Scientific Notation Addition and Subtraction in Scientific Notation • Addition and Subtraction in Scientific Notation is a little trickier. • Remember the rule for significant figures for addition and subtraction is that the least precise number sets the limit on the sig figs • When the exponents are different it is hard to tell which number is the least precise. 6 Section 2.8 Mathematical Operations in Scientific Notation Addition and Subtraction in Scientific Notation • 1.25 x 10–3 + 6.33 x 10–5 • There are two ways to deal with this • • 1. Equalize the exponents 2. Determine the uncertainty for each number and use the most uncertain (least precise) 7 Section 2.8 Mathematical Operations in Scientific Notation Addition and Subtraction in Scientific Notation • 1.25 x 10–3 + 6.33 x 10–5 • Equalize the exponents (always use the larger exponent) 1.25 x 10–3 0.0633 x 10–3 When the coefficients are added the lease precise value is 1.25 so the answer can only be precise to the 100ths place 1.25 + 0.0633 = 1.3133 = 1.31 x 10–3 • • • • 8 Section 2.8 Mathematical Operations in Scientific Notation Addition and Subtraction in Scientific Notation • 1.25 x 10–3 + 6.33 x 10–5 • • • Determine the uncertainty for each number and use the number with the larger uncertainty 1.25 x 10–3 10–2 x 10–3 = 10–5 larger uncertainty 6.33 x 10–5 10–2 x 10–5 = 10–7 • • The answer must have the same uncertainty –5 0.0013133 = 1.31 x 10–3 10–2 x 10–3 = 10–5 9 Section 2.8 Mathematical Operations in Scientific Notation Learning Check Solve the following and express the answer in scientific notation to the correct number of significant figures. 1. (7.431 x 108) + (4.00 x 106) 1. (7.431 x 10–8) – (4.00 x 10–3) 10 Section 2.8 Mathematical Operations in Scientific Notation Solution Solve the following and express the answer in scientific notation to the correct number of significant figures. (7.431 x 108) + (4.00 x 106) Equalize the exponents 7.431x 108 + 0.0400 x 108 7.4710x106 7.471x108 11 Section 2.8 Mathematical Operations in Scientific Notation Solution Solve the following and express the answer in scientific notation to the correct number of significant figures. (7.431 x 108) + (4.00 x 106) Determine the uncertainty for each number and use the larger uncertainty 7.431 x 108 10–3 x 108 = 105 larger uncertainty 4.00 x 106 10–2 x 106 = 104 The answer must have the same uncertainty 104 7.471 x 108 10–3 x 108 = 105 12 Section 2.8 Mathematical Operations in Scientific Notation Solution Solve the following and express the answer in scientific notation to the correct number of significant figures (7.431 x 10–8) – (4.00 x 10–3) Calculator answer 3.999992569 x 10 – 3 Equalize the exponents 0.00007431 x 10–3 – 4.00 x 10–3 – 4.00 x 10–3 13 Section 2.8 Mathematical Operations in Scientific Notation Solution Solve the following and express the answer in scientific notation to the correct number of significant figures. (7.431 x 10–8) – (4.00 x 10–3) Determine the uncertainty for each number and use the larger uncertainty 7.431 x 10–8 10–3 x 10–8 = 10–11 4.00 x 10–3 10–2 x 10–3 = 10–5 larger uncertainty Calculator answer 3.999992569 x 10 – 3 The answer must have the same uncertainty – 5 – 4.00 x 10–3 14