Power Operations & Scientific Notation, Lecture notes of Algebra

Download Power Operations & Scientific Notation and more Lecture notes Algebra in PDF only on Docsity! kearning, Teaching aE Student Engagement Power Operations & Scientific Notation a TheLearningCentre UNLOCK YOUR POTENTIAL 2 1. Power Operations Powers are also called exponents or indices; we can work with the indices to simplify expressions and to solve problems. Some key ideas: a) Any base number raised to the power of 1 is the base itself: for example, 51 = 5 b) Any base number raised to the power of 0 equals 1, so: 40 = 1 c) Powers can be simplified if they are multiplied or divided and have the same base. d) Powers of powers are multiplied. Hence, (23)2 = 23 × 23 = 26 e) A negative power indicates a reciprocal: 3−2 = 1 32 Certain rules apply and are often referred to as: Index Laws. Below is a summary of the index rules: Index Law Substitute variables for values 𝒂𝒂𝒎𝒎 × 𝒂𝒂𝒏𝒏 = 𝒂𝒂𝒎𝒎+𝒏𝒏 23 × 22 = 23+2 = 25 = 32 𝒂𝒂𝒎𝒎 ÷ 𝒂𝒂𝒏𝒏 = 𝒂𝒂𝒎𝒎−𝒏𝒏 36 ÷ 33 = 36−3 = 33 = 27 (𝒂𝒂𝒎𝒎)𝒏𝒏 = 𝒂𝒂𝒎𝒎𝒏𝒏 (42)5 = 42×5 = 410 = 1048 576 (𝒂𝒂𝒂𝒂)𝒎𝒎 = 𝒂𝒂𝒎𝒎𝒂𝒂𝒎𝒎 (2 × 5)2 = 22 × 52 = 4 × 25 = 100 (𝒂𝒂 𝒂𝒂� )𝒎𝒎 = 𝒂𝒂𝒎𝒎 ÷ 𝒂𝒂𝒎𝒎 (10 ÷ 5)3 = 23 = 8; (103 ÷ 53) = 1000 ÷ 125 = 8 𝒂𝒂−𝒎𝒎 = 𝟏𝟏 𝒂𝒂𝒎𝒎 4−2 = 1 42 = 1 16 𝒂𝒂 𝟏𝟏 𝒎𝒎 = √𝒂𝒂𝒎𝒎 81 3� = √83 = 2 𝒂𝒂𝟎𝟎 = 𝟏𝟏 63 ÷ 63 = 63−3 = 60 = 1; (6 ÷ 6 = 1) EXAMPLE PROBLEMS: a) Simplify 65 × 63 ÷ 62 × 72 + 64 = = 65+3−2 × 72 + 64 = 66 × 72 + 64 b) Simplify 𝑔𝑔5 × ℎ4 × 𝑔𝑔−1 = = 𝑔𝑔5 × 𝑔𝑔−1 × ℎ4 = 𝑔𝑔4 × ℎ4 Watch this short Khan Academy video for further explanation: “Simplifying expressions with exponents” https://www.khanacademy.org/math/algebra/exponent-equations/exponent-properties- algebra/v/simplifying-expressions-with-exponents 1 3. Calculations with Scientific Notation Multiplication and division calculations of quantities expressed in scientific notation follow the index laws since they all they all have the common base, i.e. base 10. Here are the steps: Multiplication Division A. Multiply the coefficients 1. Divide the coefficients B. Add their exponents 2. Subtract their exponents C. Convert the answer to scientific Notation 3. Convert the answer to scientific Notation Example: �𝟕𝟕.𝟏𝟏 × 𝟏𝟏𝟎𝟎−𝟒𝟒 � × (𝟖𝟖.𝟓𝟓 × 𝟏𝟏𝟎𝟎−𝟓𝟓 ) 𝟕𝟕.𝟏𝟏 × 𝟖𝟖.𝟓𝟓 = 𝟔𝟔𝟎𝟎.𝟑𝟑𝟓𝟓 (multiply coefficients) 𝟏𝟏𝟎𝟎−𝟒𝟒 × 𝟏𝟏𝟎𝟎−𝟓𝟓 = 𝟏𝟏𝟎𝟎(−𝟒𝟒+ −𝟓𝟓)=−𝟗𝟗 (add exponents) = 𝟔𝟔𝟎𝟎.𝟑𝟑𝟓𝟓× 𝟏𝟏𝟎𝟎−𝟗𝟗 – check it’s in scientific notation  = 𝟔𝟔.𝟎𝟎𝟑𝟑𝟓𝟓× 𝟏𝟏𝟎𝟎−𝟖𝟖 – convert to scientific notation  Example: (9 × 1020 ) ÷ (3 × 1011 ) 9 ÷ 3 = 3 (divide coefficients) 1020 ÷ 1011 = 10(20−11)=9 (subtract exponents) = 3 × 109 – check it’s in scientific notation  Recall that addition and subtraction of numbers with exponents (or indices) requires that the base and the exponent are the same. Since all numbers in scientific notation have the same base 10, for addition and subtraction calculations, we have to adjust the terms so the exponents are the same for both. This will ensure that the digits in the coefficients have the correct place value so they can be simply added or subtracted. Here are the steps: Addition Subtraction 1. Determine how much the smaller exponent must be increased by so it is equal to the larger exponent 1. Determine how much the smaller exponent must be increased by so it is equal to the larger exponent 2. Increase the smaller exponent by this number and move the decimal point of the coefficient to the left the same number of places 2. Increase the smaller exponent by this number and move the decimal point of the coefficient to the left the same number of places 3. Add the new coefficients 3. Subtract the new coefficients 4. Convert the answer to scientific notation 4. Convert the answer to scientific notation Example: �𝟑𝟑× 𝟏𝟏𝟎𝟎𝟐𝟐 �+ (𝟐𝟐 × 𝟏𝟏𝟎𝟎𝟒𝟒 ) 𝟒𝟒 − 𝟐𝟐 = 𝟐𝟐 increase the small exponent by 2 to equal the larger exponent 4 𝟎𝟎.𝟎𝟎𝟑𝟑× 𝟏𝟏𝟎𝟎𝟒𝟒 the coefficient of the first term is adjusted so its exponent matches that of the second term = �𝟎𝟎.𝟎𝟎𝟑𝟑× 𝟏𝟏𝟎𝟎𝟒𝟒 �+ �𝟐𝟐× 𝟏𝟏𝟎𝟎𝟒𝟒 � the two terms now have the same base and exponent and the coefficients can be added = 𝟐𝟐.𝟎𝟎𝟑𝟑×𝟏𝟏𝟎𝟎𝟒𝟒 check it’s in scientific notation  Example: (5.3 × 1012 ) − (4.224 × 1015 ) 15 − 12 = 3 increase the small exponent by 3 to equal the larger exponent 15 0. 0053 × 1015 the coefficient of the firs t term is adjusted so its exponent matches that of the second term = (0.0053 × 1015) − (4.224 × 1015) the two terms now have the same base and exponent and the coefficients can be subtracted. = −4.2187 × 1015 check it’s in scientific notation  2 Questions 3: a) (4.5 × 10−3 ) ÷ (3 × 102 ) b) (2.25 × 106 ) × (1.5 × 103 ) c) (6.078 × 1011 ) − (8.220 × 1014 ) (give answer to 4 significant figures). d) (3.67 × 105 ) × (23.6 × 104 ) e) (7.6 × 10−3 ) + (√9.0 × 10−2) f) Two particles weigh 2.43 X 10-2 grams and 3.04 X 10-3 grams. What is the difference in their weight in scientific notation? g) How long does it take light to travel to the Earth from the Sun in seconds, given that the Earth is 1.5 X 108 km from the Sun and the speed of light is 3 X 105 km/s? 3 4 Answers Q1. Power Operations a) i. 52 × 54 + 52 = 56 + 52 ii. x2 × x5 = x7 iii. 42 × t3 ÷ 42 = t3 iv. (54)3 = 512 v. 2436 34 = 2432 = 16 × 9 = 144 vi. 32 × 3−5 = 3−3 = 1 27 vii. 9(𝑥𝑥2)3 3𝑥𝑥𝑥𝑥2 = 9𝑥𝑥6 3𝑥𝑥𝑥𝑥2 = 3𝑥𝑥 5 𝑥𝑥2 viii. 𝑎𝑎−1√𝑎𝑎 = 𝑎𝑎−1 × 𝑎𝑎 1 2 = 𝑎𝑎− 1 2 = 1 √𝑎𝑎 𝑜𝑜𝑜𝑜 1 𝑎𝑎 1 2 b) i. 𝑥𝑥 = 2 ii. 𝑥𝑥 = −2 iii. 𝑥𝑥 = 3 iv. 𝑥𝑥 = 5 v. Show that 16𝑎𝑎 2𝑏𝑏3 3𝑎𝑎3𝑏𝑏 ÷ 8𝑏𝑏 2𝑎𝑎 9𝑎𝑎3𝑏𝑏5 = 6𝑎𝑎𝑏𝑏5 16𝑎𝑎 2𝑏𝑏3 3𝑎𝑎3𝑏𝑏 × 9𝑎𝑎3𝑏𝑏5 8𝑏𝑏2𝑎𝑎 = 2𝑎𝑎 5𝑏𝑏8×3 𝑏𝑏3𝑎𝑎4 = 2𝑎𝑎 1𝑏𝑏5×3 1 = 6𝑎𝑎𝑏𝑏5 Q2. Scientific Notation a) 4.5 x 102 b) 9.0 x 107 c) 3.5 x 100 d) 9.75 x 10-2 e) 375 f) 39.7 g) 0.1875 h) 0.00875 Q3. Calculations with scientific notation a) 1.5 × 10−5 b) 3.375 × 109 c) −8.214 × 1014 d) 8.8612 × 1010 e) 3.076 × 10−2 f) 2.126 × 10−2 g) 500 s