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Algebra II Applications of Powers Unit Plan Now that weβve learned the properties of powers, we can put those properties to use in some real-life applications. In this section we will be examining the following topics: - Scientific Notation Solving Equations that involve Powers Numerical Equations that Involve Powers Falling Object Model Pythagorean Theorem Other Geometry Applications I. Scientific Notation Scientific Notation is a way of expressing the values of very large or very small quantities without filling the board (or the universe!) with zeros. A number is in scientific notation if it is written in the form π β 10π , with certain restrictions on βaβ and βbβ Definitions: Mantissa β The name for βa.β It must be a number greater than or equal to one and strictly less than 10. Magnitude β The name for βb.β A positive magnitude represents a large number (bigger than 10). A negative magnitude represents a small number (smaller than 1.) Decimal form β What we call βordinaryβ numbers in contrast to scientific notation numbers. Key Idea: Moving the decimal point left adds to the magnitude. Moving it to the right subtracts from the magnitude. 1.) Converting a number in scientific notation to decimal (standard) form Ex: 5.21 β 107 = 52,100,000 Ex: 4.8 β 10β4 = 0.0004 Leading Zero β For small numbers it is advised to put a leading zero before the decimal point, because it is easy for the decimal point to get lost, and because this makes the number of zeroes equal the absolute value of the magnitude. 2.) Converting a number in decimal (standard) form to scientific notation Key Idea: Start with a hidden β10^0β Ex: 45,000 = 45,000 β 100 = 4.5 β 104 HW: p. 751 #1-55 odds (evens if necessary) Ex: 0.000083 = 0.000083 β 100 = 8.3 β 10β5 3.) Operations with Scientific Notation Key Idea: Each scientific notation number must have its own set of parentheses. Ex: What is 3.8 β 1012 divided by 9.4 β 10β6 ? [Answer: 4.04 β 1017 ] Ex: A cube has a side length of 9.23 β 1012 parsecs. What is the volume of this cube? [Answer: 7.86 β 1038 parsecs] What about numbers too big for your calculator? HW: p. 299 #36-39 4.) Large and Small Number Names Key Idea: All the powers of 10 from 0 to 3 have their own number names. After that, every third power gets a number name. One: 100 Ten: 101 Tenth: 10β1 Hundred: 102 Hundredth: 10β2 Thousand: 103 Thousandth: 10β3 Million: 106 Millionth: 10β6 Billion: 109 Trillion: 1012 Billionth: 10β9 Trillionth: 10β12 Ex: How would we express 52 billion in scientific notation? [Answer: 5.2 β 1010 ] Ex: How would we express fifteen hundredths in scientific notation? [Answer: 1.5 β 10β1 ] 5.) Scientific Notation Word Problems Key Idea: If you are unsure of what operation to use, imagine smaller numbers. Then, apply the correct operation, keeping in mind that every scientific notation number gets its own set of parentheses! Ex: There are 50 trillion cells in the human body. There are approximately one thousand people in a school. How many human cells are in this school? Express your answer in scientific notation. (5.0 β 1013 ) β (1.0 β 103 ) = 5.0 β 1016 Ex: The output of the US Economy is about 17 trillion dollars per year. There are about 300 million people in the US. What is the average economic output per person in the US? Round to the nearest thousand dollars. (1.7 β 1013 ) β $57,000 (3.0 β 108 ) HW: p. 300-301 #45-53 6.) Shortcuts in your calculator Key Idea: Your (blue) calculators have a βSCIβ mode that makes every answer appear in scientific notation. To access it, press β2ndβ then βDRGβ. Put the cursor under βSCIβ and hit enter. HW: Scientific Notation Topic Practice Scientific Notation Quiz II. Solving Numerical Equations that involve Powers 1.) Perfect Squares β A perfect square is a number that is a square of a whole number. These will become important in this section. Ex: Name the first eleven perfect squares [Answer: 0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 ] 2.) Roots β When the base is unknown, the opposite of power is root. The most common root is square root. When there is no number in the crook of the root symbol, the root is assumed to be a square root. Key Idea: The calculator tells you there is only one output for root, but even roots have two outputs, one positive and one negative. We can represent this with the symbol ± Ex: Solve for x: 3π₯ 2 β 8 = 19 [ Answer: π₯ = ±3 ] Ex: Solve for x: (π₯ + 4)2 = 100 [ Answer: π₯ = { β6, 14} ] 3.) Simplified Root Form β Often, when the answer to a problem like those above is a decimal, we do not write the decimal, but rather use something called simplified root form (SRF). To use simplified root form, follow these steps: 1.) Find the largest perfect square factor of the number under the root. (Assuming the root is a square root.) Ex: β150 = β25 β 6 2.) Re-write the expression as two separate roots (The Product Property of Powers allows us to do this since βπ₯ = π₯ 1/2 .) Ex: β25 β 6 = β25 β β6 3.) Evaluate the root that has the perfect square under it, but not the other root. Ex: β25 β β6 = ±5β6 Technically the ± is implied by the continued presence of the β symbol in the expression, but tradition dictates that we write it out anyway. Ex: Re-write β72 in simplified root form. [ Answer: ±6β2 ] Ex: Re-write β12 in simplified root form. [ Answer: ±2β3 ] HW: p. 258 #3-5, #18-23; p. 260 #75; p. 266 #10 4.) Solving Equations Using Simplified Root Form Ex: Solve the equation π₯ 2 + 5 = 13 . Put your answer in simplified root form. [Answer: π₯ = ±2β2 ] Ex: Solve the equation (π₯ β 7)2 = 48 . Put your answer in simplified root form [ Answer: π₯ = 7 ± 4β3 ] HW: p. 257 #4-6; p. 289 #16-18; p.258-260 #12-17; #46-68; #77-78; p. 782 #27 - 30 p. 266 #16-21; p. 287 #20 β 23; 5.) Solving Equations Using Higher Roots β Not all problems that involve exponents have an exponent of β2β. Many equations must be solved using βhigherβ roots. When a higher root is used, the value of the exponent that is being eliminated is put in the βcrookβ of the root. Otherwise, it is assumed that we are dealing with square roots. Key Idea: Odd roots are functions, so there is only one output. The odd root of a negative number is negative, and the odd root of a positive number is positive. Ex: Solve for x: π₯ 3 = 8 [ Answer: π₯ = 2 ] [Note how β 2 would not work as a solution here because ( β2)3 = β8 ] Ex: Solve for x: π₯ 4 = 16 [ Answer: π₯ = ±2 ] [Note how the ± sign is used here, because the exponent is even.] Key Idea: To access the higher roots in your calculator, press the value of the root that you want, then the β2ndβ button, then the β^β button, and then the number you want under the root. Remember, in the case of even roots, there is still a negative output that your calculator canβt see! Ex: Solve for x: π₯ 5 = 60 [ Answer: π₯ β 2.27 ] HW: p. 356 #36-43 first, then #6-8, 27-35; p. 401 #5-8; p. 784 #1-5 6.) Falling Object Model β This physics equation models the height of a falling object as a function of its starting height and the time. There are many different ways to write the Falling Object Model, we will choose a simple way, which is β = β16π‘ 2 + π , where β is the current height, π‘ is time and π Book work: p. 257 example 4; #7; p. 259 #69-73; p. 289 #39 7. Pythagorean Theorem β The Pythagorean Theorem relates the lengths of the sides of a right triangle. The sides that form the right angle are called the legs, and the third side, which does not touch the right angle is called the hypotenuse. The hypotenuse is always the longest side of the triangle. The legs are assigned the variables π and π , and the hypotenuse is given the variable π . The equation of the Theorem is π2 + π 2 = π 2 . Ex: A right triangle has legs of three and four inches. What is its hypotenuse? [Answer: 5 inches.] Ex: A right triangle has a leg of 5 inches and a hypotenuse of 13 inches. What is the length of the other leg? [Answer: 12 inches.] Ex: A right triangle is isosceles (meaning the two legs are the same length.) Its hypotenuse is 20 cm. What is the length of each of the legs? [Answer: about 14.1 inches.] The converse of this Theorem is also true, that is, if a triangle has three sides, and you substitute their lengths into the formula π2 + π 2 = π 2 so that the longest side is assigned to π , then you have proven that this triangle is a right triangle. Ex: A triangle has sides of 10, 13 and 16 inches. Is this triangle a right triangle? [Answer : No.] The Pythagorean Theorem can also be used to find the distance between two points on the coordinate plane. Ex: What is the distance between (4, 7) and (2, 3)? [Answer: About 4.47] When we are using the Theorem to find distance, it is often re-written as (βπ₯)2 + (βπ¦)2 = π2 HW (distance formula): p. 691 #1-6; p. 694 #3-11 HW: Pythagorean Theorem Topic Practice Pythagorean Theorem Quiz? 8.) Other Geometry Applications β Roots are useful to find the unknowns in a number of geometry formulas. Ex: A circle has an area of 100 square inches. Approximate its radius to the nearest tenth of an inch. [Answer: 5.6 inches.] Ex: A sphere has a volume of 100 cubic inches. Approximate its radius to the nearest tenth of an inch. [Answer: 2.9 inches] Ex: A rectangular prism has a width that is twice its height and a length that is five times its height. The volume of this prism is 5,120 cubic inches. What are the dimensions of this prism? [Answer: 8 by 16 by 40 inches.] HW: p. 260 #97-99; p. 357 #84,85 AOP Worksheets & Blockbuster: AOP Test