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Section 5.1 Exponents and Scientific Notation

Exponents that are natural numbers are shorthand notation for repeating factors. 3 4 = is the base 4 is the exponent (also called power) Note, by the order of operations, exponents are calculated before all other operations, except expressions in parentheses or other grouping symbols.

Product Rule (applies to common bases only) a m a n = a m+n Simplify each of the following expressions. Example = 3 6 = = 729 x 4 x 5 = x 4+5 z 3 z 2 z 5 = z (3y 2 )(-4y 4 )= 3 y 2 -4 y 4 = (3 -4)(y 2 y 4 )= -12y 6 = = x 9 = z 10

Zero exponent a 0 = 1, a  0 Note: 0 0 is undefined. Simplify each of the following expressions. 5 0 Example = 1 (xyz 3 ) 0 = x 0 y 0 (z 3 ) 0 = = 1 -x0-x0 = -1∙x 0 = -1 ∙1 = -1 (Assume all variables have nonzero values.)

Problem from today’s homework:

Quotient Rule (applies to common bases only) Example Simplify the following expression. Group common bases together

Problem from today’s homework:

Using the quotient rule, But what does x -2 mean?

In order to extend the quotient rule to cases where the difference of the exponents would give us a negative number we define negative exponents as follows: If a  0, and n is an integer, then

Simplify by writing each of the following expressions with positive exponents or calculating. Example Remember that without parentheses, only the x is the base for the exponent –4, not the entire expression 2x. 1)2)3)

Note that in the previous problem, 2x -4 gave us 2 on top and x 4 on the bottom, and only the x term, not the 2, was raised to the -4th power. Here’s a question for you: What would (2x) -4 look like when simplified? Would it look different than 2x -4 ? (Good question for a quiz!)

Example Simplify by writing each of the following expressions with positive exponents or calculating. 1) 2) 3) Notice the difference in results when parentheses enclose the -3

Example Simplify by writing each of the following expressions with positive exponents. 1) 2) (Note that to convert a power with a negative exponent to one with a positive exponent, you simply switch the power from a numerator to a denominator, or vice versa, and switch the exponent to its positive value.)

Problem from today’s homework:

In many fields of science we encounter very large or very small numbers. Scientific notation is a convenient shorthand for expressing these types of numbers using powers of the base 10. A positive number is written in scientific notation if it is written as a product of a number a, where 1  a < 10, and an integer power r of 10. a  10 r

To Write a Number in Scientific Notation 1)Move the decimal point in the original number to the left or right, so that the new number has a value between 1 and 10. 2)Count the number of decimal places the decimal point is moved in Step 1. If the original number is 10 or greater, the count is positive. If the original number is less than 1, the count is negative. 3)Multiply the new number in Step 1 by 10 raised to an exponent equal to the count found in Step 2.

Example Write each of the following in scientific notation ) You must move the decimal 3 places to the left, so that the new number has a value between 1 and 10. Since we moved the decimal 3 places, and the original number was > 10, our count is positive = 4.7  ) Have to move the decimal 4 places to the right, so that the new number has a value between 1 and 10. Since we moved the decimal 4 places, and the original number was < 1, our count is negative = 4.7  10 -4

To Write a Scientific Notation Number in Standard Form Move the decimal point the same number of spaces as the exponent on 10. If the exponent is positive, move the decimal point to the right. If the exponent is negative, move the decimal point to the left.

Example Write each of the following in standard notation  ) Since the exponent is a positive 3, we move the decimal 3 places to the right  10 3 =  ) Since the exponent is a negative 5, we move the decimal 5 places to the left  =

Reminder: This homework assignment on section 5.1 is due at the start of next class period. Note: There are 48 problems in this assignment, but most of them are short. (This assignment took last semester’s students just over an hour to complete.)

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