Pre-Algebra 8 Notes Exponents and Scientific Notation

Transcription

1 Pre-Algebra 8 Notes Eponents and Scientific Notation Rules of Eponents CCSS 8.EE.A.: Know and apply the properties of integer eponents to generate equivalent numerical epressions. Review with students that an eponent is the superscript which tells how many times the base is used as a factor. eponent base In the number, read to the third power or cubed, the is called the base and the is called the eponent. Eamples: 4 To write an eponential in standard form, compute the products. i.e. Since we will be addressing powers of 0 in scientific notation, emphasize this in your eamples. Eamples: What pattern allows you to find the value of an eponential with base 0 quickly? Answer: The number of zeroes is equal to the eponent! Caution: If a number does not have an eponent visible, it is understood to have an eponent of ONE! Let s practice writing numbers in eponential form. Pre-Algebra 8, Unit: Eponents and Scientific Notation Page of 4 Revised 0 - CCSS

2 Eamples: Write 8 with a base of. 8, 8, therefore 8? 4 Write with a base of.,, therefore? In algebra, we often have to find the products and quotients of algebraic epression. For eample, what is the product of the problem below?? Caution: Many students will jump to an answer of, which is incorrect. Watch for this error! Have students rewrite each term in epanded form, and then convert it back to eponential form. Since and, ( ) ( ) or. We do not multiply the eponents as we might suspect: we add them! Let s try a few more problems to verify our conjecture: Eamples: ( ) ( ) or 4 ( ) or ie, or 4 4+ ie, or + We are now ready to state the rule for multiplying eponential epressions with the same base. When multiplying powers with the same base, add their eponents; a b a b that is, +. What might we suspect about the rule for division? Since division is the inverse of multiplication, and multiplying eponential epressions involves the addition of eponents, what would division of eponential epressions involve? (Note how Pre-Algebra 8, Unit: Eponents and Scientific Notation Page of 4 Revised 0 - CCSS

3 confusing this all seems!) We might suggest subtraction is the key here; we can show this to be true with a few eamples: Eample: ie, or Eample: Let s now state the rule: ie, or When dividing powers with the same base, subtract their eponents (subtract the eponent in the denominator from the eponent in the numerator); a a b that is, b Emphasize with students to be careful with their integer operations now they may have the tendency to add when they should multiply. A simple eample is to look at the following 4 product: 4. The answer is 8, of course. Common errors are to multiply all numbers involved, arriving at the incorrect answer of 8 ; or to add all numbers, arriving at the incorrect answer of. Look at the following eample for other errors to watch for. Eample: Eample: 8 4 subtractingeverything: 4 ; but watch for answersof dividingeverything: a a a ; but watch for answersof adding everything: 8a multiplying everything: a 0 At this point, students will probably be feeling that math rules do not always make sense! Emphasize that they can always go back to epanding the epression to notation without eponents to arrive at the answer. To see what happens when you raise a power to a power, use the order of operations. Pre-Algebra 8, Unit: Eponents and Scientific Notation Page of 4 Revised 0 - CCSS

4 ( ) ( ) ( )( ) Evaluate the power inside the parentheses. Evaluate the power outside the parentheses. When raising a power to a power, keep the base and multiply the eponents; that is, ( a) b ab Eamples: ( ) 0 0( ) ( ) 0 ( ) ( ) 4 Negative and Zero Eponents Pattern development is a very effective way to introduce the concept of negative and zero eponents. Consider the following pattern that students should have seen previously As we review this pattern, students should see that each time the eponent is decreased by, the epanded form contains one less factor of and the product is half of the preceding product. Pre-Algebra 8, Unit: Eponents and Scientific Notation Page 4 of 4 Revised 0 - CCSS

5 So, and 4. Following this pattern, is. Continuing this pattern, 4 Looking at powers of 0, or or ,000 or Applying Eponents: Scientific Notation CCSS 8.EE. A.: Use numbers epressed in the form of a single digit times an integer power of 0 to estimate very large or very small quantities, and to epress how many times as much one is than the other. n A number is written in scientific notation if it has the form 0 where < 0 and n is an integer. An eample of a number written in scientific notation is To convert a number written in scientific notation to standard form, the eponent tells you how many times to move the decimal point to the right or left. Pre-Algebra 8, Unit: Eponents and Scientific Notation Page of 4 Revised 0 - CCSS