P.2 Properties of exponents Simplifying exponential expressions Converting from Decimal Notation to Scientific Notation Converting from Scientific Notation to Decimal Notation Pg. 25 #26-86 every other even Note: For the simplifying exponential expressions section, leave your answers with all positive exponents in them – no negative exponents!

The Negative Exponent Rule: When an exponent is negative, the entire power (base and exponent) Examples: can be moved to the other side of the fraction. Once moved, the exponent is written as a positive. Simplify the power to finish the problem. The Zero Exponent Rule: Anything raised to the zero power becomes the value 1. Examples: (-3) 0 = 1 (6x 4 y -3 ) 0 = 1

Simplifying by Grouping Same Base Powers 4 5 ∙4 2 = 4∙4∙4∙4∙4 times 4∙4 A group of five 4s times a group of two 4s is: 4∙4∙4∙4∙4∙4∙4 = 4 7 (3x 2 y 4 ) 5 = (3x 2 y 4 ) (3x 2 y 4 ) (3x 2 y 4 ) (3x 2 y 4 ) (3x 2 y 4 ) Five groups of (3x 2 y 4 ) = 3 ∙ 3 ∙ 3 ∙ 3 ∙ 3 ∙ x 2 ∙ x 2 ∙ x 2 ∙ x 2 ∙ x 2 ∙ y 4 ∙ y 4∙ y 4 ∙ y 4∙ y 4 = 243 x 10 y 20

M 5 = M∙M∙M∙M∙M = M∙M∙M∙M∙M = M∙M = M 2 M 3 M∙M∙M M∙M∙M M = 1 These M pairs cancel; leaving only two Ms in the numerator. M ( 2 ∕ 7 ) 3 = (two sevenths raised to the third power) = ( 2 ∕ 7 )( 2 ∕ 7 )( 2 ∕ 7 ) 2∙2∙2 = 8 7∙7∙7 343

Simplify: A)-35x 2 y 3 = 5x 6 y -8 Divide -35 by 5 = -7 = -7x 2 y 3 x 6 y -8 = x∙x Two of the x pairs cancel. x∙x∙x∙x∙x∙x = -7y 3 x 4 y -8 The y -8 must move to the numerator and the exponent becomes positive 8. = -7y 3 y 8 x 4 Finally, exponential groups of base y in the numerator can be combined. = -7y 11 x 4

Simplify: 1.(2x 3 y 2 ) 4 2.(-6x 2 y 5 )(3xy 3 )

Your final simplified expression should have no negative exponents

In Decimal Notation (how number values are most commonly written), very large and very small numbers require a long series of zeros to denote. For example: and 1,367,000,000,000 Scientific Notation was invented so that numbers of very large or very small size could be written with fewer symbols. Here are the above numbers in scientific notation: 2.37 x and x Notice that the very small number on the left ( ) is associated with a negative exponential power of ten in its scientific notation (-9). Conversely, the very large number on the right (1,367,000,000,000 ) is associated with a positive exponential power of ten in its scientific notation (12). This correlation always holds true in these conversions.

To Convert from Decimal (Normal) Notation to Scientific Notation: A) Take the decimal and move it to the right of the first non-zero digit you pass. Like this: You moved it seven places right to obtain the number 4.08 Since the value itself ( ) is actually very small, we use negative exponent in our final answer: 4.08 x B) 120,500 Take the decimal (which is unseen on the far right) and move it to the left until there is only one non-zero digit left. Like this becomes You moved it five places to the left to obtain the number Since the number itself (120,500) is actually very large, we use a positive exponent in our final answer: x 10 5

To Convert from Scientific Notation to Decimal (Normal) Notation: C)7.035 x 10 6 This exponent is positive (6), which means that the actual number is very large. So, we should move the decimal to the RIGHT to make the value larger. The exponent tells us how many places right we should go: six Starting with and moving right six places we get 7 035_ _ _. Moving the decimal in this way leaves us with empty spaces. We fill them in with zeros to obtain our final answer: Or, if you prefer: 7,035,000 D) 8.65 x This exponent is negative (-2), which means the actual number is very small. So, we should move the decimal to the LEFT to make the value smaller. The exponent tells us how many places left we should go: two Starting with 8.65 and moving left two places we get. _ 865 Again, moving the decimal in this way leaves us with empty spaces. Again, we fill them in with zeros to obtain our final answer:.0865 Or, if you prefer:

Write each number in decimal notation x x Write each number in scientific notation. 7.5,210,000,