Section 3.1 - Scientific Notation Scientific notation is used to express very large and very small numbers. Powers of 10 are used in scientific notation. Fill in the table and look for patterns. Meaning Standard Notation 10 4 10 3 10 2 10 1 10 0 10-1 10-2 10-3 10 n, when n is positive, is a whole number with a 1 followed by zeros. 10 n, when n is negative, is a number between 0 and 1. One is the last digit and the n tells how many places are. Scientific Notation of a POSITIVE value consists of a number between 1 and 10 that is multiplied by a power of 10. A number is written in scientific notation when it is in the form, n 10 y. n always has nonzero digit in front of the decimal point and y tells how many places and the direction that you moved the point. Unnecessary zeros are omitted from n. Change each number from standard notation to scientific notation. 360,000 2,040,000,000 The Blue-ray disc has a capacity of 50,000,000,000 bytes. 1
Change each number given in scientific notation to standard notation 5.1 x 10 6 6.02 x 10 8 The original ipod had a capacity of 5 GB, that is 5 x 10 9 bytes. If the number is greater than ZERO but less than ONE, then 1 n 10 and y will be an integer less than 0. Change each number from standard notation to scientific notation. 0.28 0.0000504 The typical thickness of bond paper is 0.0038 inches. Change each number from scientific notation to standard notation. 3.6 x 10-5 4.105 x 10-11 The diameter of the salmonella bacteria is typically 5 x 10-5 cm. Number line summary for all positive values, n x 10 y : 0 1 Multiplying Powers with the Same Base or the Product Rule of Exponents: a m a n = a m+n When you multiply two numbers with the same keep the and add the Examples 4 2 4 3 10 2 10 10 3 x 3 x 5 x 10-3 10 5 10-4 10-2 2
When multiplying numbers written in scientific notation, rearrange the factors and multiply similar numbers. Express your final answer in both scientific notation and standard notation. 8 4 (4.2 10 )(3.37 10 ) 4.2 3.37 10 10 8 4 2.75 10 3 7 3.1 10 5 2 2.05 10 5.7 10 The typical thickness of a Pringles potato chip is 0.058 inches. If there are 2952 chips in a stack, how tall is the stack of chips? Express your answer in both scientific notation and standard notation. Section 3.2 Using a Scientific Calculator in Algebra Since scientific and graphing calculators are programmed to follow order of operations, you can often enter an expression into your calculator the way it is written on paper. Keys to know: subtraction key, usually grouped with the keys for multiplication and division ( ) or +/- negative number Per form the following operations by hand. Then check your result with your calculator. -8 + 4 By hand In your calculator -8 4 By hand In your calculator -8 4 By hand In your calculator 8 4 By hand In your calculator 3
If your results by hand and calculator are the same, continue with the next group of problems. Be sure to show your steps in solving each problem by hand. ( or ) or [( or )] Parentheses are often located just above the number keys -9 (-9) By hand In your calculator 12 (-3) By hand In your calculator -5 + 6 (-7) By hand In your calculator 7 (-4) (-1) By hand In your calculator (-4)(-7)(-2) By hand In your calculator 8(-2)(-6) By hand In your calculator 5 6 3 By hand In your calculator 8 + 16 4 By hand In your calculator 2 9(3 4) By hand In your calculator 4 + 7(-8 + 2) By hand In your calculator ^ or y x or x y Exponent keys used for powers. X 2 special key on some calculators for squaring a number Show your work by hand for each problem, and then enter it into your calculator. 10 4 By hand In your calculator (-3) 4 By hand In your calculator -3 4 By hand In your calculator 10-3 By hand In your calculator Remember the division bar is a grouping symbol and you must use parentheses around any numerator or denominator that has an operation in it. (-7 + 3)(-2 8) By hand In your calculator 4
-5 7(5-3 2 ) By hand In your calculator 4 3 2 5 2 By hand In your calculator 8 3(5 3) 2 By hand In your calculator 2 3 4 6 4 2 1 By hand In your calculator 3 3 5 20 By hand In your calculator 5 4 8 5 3 3 1 12 10 4 6 By hand In your calculator Scientific Notation on your calculator EE or Exp keys are used to enter numbers in scientific notation. You many need to use a shift key and the base 10 may not appear as you type the number. See page 226 for common ways that scientific calculators display scientific notation. We need scientific notation to solve problems such as (9,000,000,000)(20,000,000)and (0.00009)(.000000005) on our calculators. First rewrite each number in scientific notation, and then compute by hand and with your calculator. (9,000,000,000)(20,000,000) = (0.00009)(.000000005) = 5
Perform each of the following calculations on your calculator. Copy the answer on your display screen and then write your answer in scientific notation. 5.8 10 5 Calculator answer Scientific Notation 8 10-6 Calculator answer Scientific Notation (2.5 10 8 )(1.7 10 11 ) Calculator answer Scientific Notation (5.7 10-7 )(2.3 10 12 ) Calculator answer Scientific Notation Section 3.3 - The Distributive Property and Like Terms 2( 2 + 3) = 2 2 + 2 3 When we have numbers inside parentheses we use order of operations to simplify, but if we have a variable(s) inside the parentheses, we use the property. For all real numbers a, b, and c, a ( b + c) = ab + ac or (b + c) a = ab + ac a ( b - c) = ab ac or (b c) a = ab ac Use the distributive property to rewrite the following algebraic expressions without parentheses. Remember distribute or the multiplication over the sum or difference to the expression. 3 (2x + 1) -4 (3y + 5) -3 (2x 7) 6
- (4x + 5) - (7 2x) (-3x + 5) 6 Combining Like Terms Two terms are like terms or similar terms if the parts of the two terms are exactly the same, including the exponents. Like terms Not like terms When we can add or subtract like terms. We CANNOT add or subtract unlike terms. Completely simplify each of the following expressions by combining like terms, where possible. Remember that when we combine like terms only the changes. The remains the same. 8m + 2m 3m + 1-5f + 2 2f + 3 7x 2 6x 2 + 5x 7y 10 x 2 5 Completely simplify the following expressions by applying the distributive property and then combining like terms. 4y 2 (3y 8) -3n 2 (6 + 2n) n + n 7(n 2) 8 3(12x 4) 2x 4(3x + 1) (2x 1) -(2x 5) (-3x) 7
Section 3.4 Equations with Like Terms Steps: 1) Simplify each side of the equation, if you can. 2) Undo what has been done to the variable in the reverse order of order of operations. Solve each equation. Check your solution -4n + 10n + 18 = 18 15n 4 4(3n) = -7 6r 3 2r = 9 7y 12 3y = 18 Using the Distributive Property to Solve Equations Steps: 1) Simplify each side of the equation. First distribute; then combine like terms. 2) Undo what has been done to the variable in the reverse order of order of operations. -4(y + 5) + 2y = -4 Solve and check each equation. 3 (x 2) = -15 5 3(x 4) = 19 8
-9 2n + 2 (9 + 4n) = -21-17t 4 (2t + 1) = 96 Writing and Solving Equations 1) Choose a letter to represent the variable quantity variable. 2) Translate the problem into an equation. 3) Check the solution in two ways. The product of 3 and the difference of a number and 2 is 21. Find the number. Bob math class has 34 students enrolled. If there are 4 more boys than girls in his class, how many boys and how many girls are in his math class? Sally spent $32 on a board game and a puzzle. If the puzzle cost twelve dollars less than the board game, how much did she pay for each? 9