2. © William James Calhoun, 2001 Warm up… Simplify 1. 2. The number of three-letter “words” that can be formed with the English alphabet is 26 3. the number of five-letter “words” that can be formed 26 5. How many times more five-letter “words” can be formed that three-letter words?

3. © William James Calhoun, 2001 Scientific Notation

4. © William James Calhoun, 2001 OBJECTIVES: You will simplify expressions involving quotients of monomials and simplify expressions containing negative exponents.

5. © William James Calhoun, 2001 In science, very small numbers are found when measuring energy levels of electrons and atomic radii. Also in science, extremely large numbers are found when measuring distances between stars or energy levels of nuclear explosions. Extremely small numbers have lots of decimal places with most of those decimals being zeros. Extremely large numbers have lots of zero place holders. People working with such numbers wanted to find an easier way to write these numbers without having to put in all those zeros.

6. © William James Calhoun, 2001 Enter: Scientific Notation. Scientific Notation is a way to write either very large or very small numbers without all the zeros needed for place-holding. It can also be used to write any numbers. To write a number in scientific notation, you must first get the decimal place behind the very first digit of the number. Count how many places you moved the decimal. Add “x10” to the end of your number and raise the “x10” to a power based on the number of places the decimal moved. The “x10 power ” alleviates the zeros.

7. © William James Calhoun, 2001 For example, there are 29,000,000 passengers that arrive and depart from Miami International Airport. That number has a lot of zeros. Scientific Notation is helpful in this case to write 29 million in a more compact form. The decimal needs to be behind the “2” - the first digit of the number. It is currently behind the last zero. 29,000,000.. The decimal moved seven places to the left. Written in scientific notation, 29 million is 2.9 x10 7. A number is expressed in scientific notation when it is in the form a x 10 n, where 1  a < 10 and n is an integer. DEFINITION OF SCIENTIFIC NOTATION

8. © William James Calhoun, 2001 With very small numbers in scientific notation, the exponent on the x10 will be a negative number. Knowing whether to make the exponent on the x10 positive or negative is the hardest part of scientific notation for most students. You need to remember: Positive x10 means the number is bigger written out the long way. So there usually needs to be some zeros as place holders at the end of the number. 8.3 x10 5 = 830,000 Negative x10 means the number is smaller written out the long way. So there usually needs to be some zeros as place holders between the decimal and the first non-zero number. 8.3 x10 -5 = 0.000083

9. © William James Calhoun, 2001 EXAMPLE 1: Express each number in scientific notation. A. 98,700,000,000B. 0.0000056 Move the decimal. Remember the decimal is hidden at the end. The decimal moved how many places? 10 The number is larger than its scientific notation, so the exponent will be… positive. The answer is then: 9.8 x10 10 Move the decimal. The decimal moved how many places? 6 The number is larger than its scientific notation, so the exponent will be… negative. The answer is then: 5.6 x10 -6 The book shows a different way of doing this example. Use that method if you wish. I prefer my way of doing scientific notation.

10. © William James Calhoun, 2001 You try… Express each number in scientific notation. 1.915,600,000,000 2.0.003157

11. © William James Calhoun, 2001 EXAMPLE 2: Express each number in standard notation. A. 3.54 x10 5 B. 9.72 x10 -4 The power is positive, so the number in standard should be larger than in scientific notation. To make a larger number, we need to move the decimal to the… right. How many places? 5 Do so. 3.54 The power is negative, so the number in standard should be smaller than in scientific notation. To make a smaller number, we need to move the decimal to the… left. How many places? 4 Do so. 354,000 9.72 Insert place holders between the decimal and the next nearest digit. 000 Re-write putting a zero in front to notice the decimal point. 0.000972 Insert place holders between the decimal and the next nearest digit.

12. © William James Calhoun, 2001 EXAMPLE 3: Use scientific notation to evaluate each expression. A. (610)(2,500,000,000)B. (0.000009)(3700) Some calculators can handle really big/small numbers. Others cannot. For the others that cannot, there is a way to use scientific notation to get answers. Also, most calculators nowadays can input scientific notation using the “x10” key or the “ EE ” key. First, we will rewrite both problems in scientific notation. A. (6.1 x10 2 )(2.5 x10 9 )A. (9 x10 -6 )(3.7 x10 3 ) Now, the “x10” is the same base in both parts. Also, we are dealing with multiplication. When you multiply a same base, you simply… add the exponents. So, multiply the number parts together then add the exponents on the x10. (6.1 x 2.5)(10 2 x 10 9 ) 15.25 x10 11 (9 x 3.7)(10 -6 x 10 3 ) 33.3 x10 -3 Get these two answers back into scientific notation by move the decimal over on place to the left. 1.525 x10 12 = 1,525,000,000,000 3.33 x10 -2 = 0.0333

13. © William James Calhoun, 2001 EXAMPLE 4: Use scientific notation to evaluate the division expression: First, divide the number in the top by the number in the bottom, ignoring the x10 notation. Take care of the x10 notation. = x10 (8 - 3) = x10 5 Rewrite, then change the answer so the decimal is behind the first digit. 0.644 x10 5 = 6.44 x10 4.

14. © William James Calhoun, 2001 You try… Evaluate. Express each result in scientific and standard notation. 1.2.

15. © William James Calhoun, 2001 CLASS WORK Page 428 #16-17, 18-57 multiples of 3 And number 58