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Chapter 1 Section 8 Exponential Notation and Order of Operations Definition An exponent is used to indicate repeated multiplication. The exponent tells how many times the base is used as a factor. Example : 23 = 2 ⋅ 2 ⋅ 2 2 is the base, 3 is the exponent Exponential Notation For any natural number n, n factors n b = b ⋅ b ⋅ b ⋅ b ⋅ ... ⋅ b 1 Rewrite the following without exponents. Simplify where appropriate 34 x2 ( −2 ) 4 − x2 2 (−x ) −24 Evaluate the Following 2 ( ) For x = 9: (a) −x ; (b) − x 2 ( For x = 9 and y = 3: x + y 2 ) Bob said Paul is tall. Bob, said Paul, is tall 2 2 + 3⋅5 Order of Operations Agreement PEMDAS • • • • • • • Parentheses or absolute values first Exponents Multiplication or Division Addition or Subtraction Similar operations are done from left to right. Order of Operations Agreement • Remove parentheses or absolute values, work from the innermost symbols first. • Exponents • Multiplication or Division • Addition or Subtraction • Similar operations are done from left to right. 3 When solving problems with a fraction bar, simplify the numerator and denominator first, then do the division last. The fraction bar acts as a set of parentheses For example: 3(14) − 4 23 − 6 Equivalent Notation x −1 = (x − 1) / 2 2 x2 − 4 = x2 − 4 / x − 2 x−2 ( )( ) Your Turn 10 + 5 6 −1 52 + 17 2 ⋅ 5 − 23 (5 − 3)2 + 2 42 − (8 + 2) 4 Definition Two numbers that are the same distance away from zero on a number line, but on opposite sides of zero, are opposites or negatives. Definition If a is any number, the opposite of a is denoted by -a (read the opposite of a) Example Find the opposite of each number • • • • • • -4 3 9 -5 a x 5 The Double Negative Rule If a is any number, -(-a) = a Example Simplify each of the following. • • • • • • -(-8) = -(-4) = -(-7) = -(-s) = -(-u) = -(-(-3)) = Definition The absolute value of any number is the distance between the number and zero on a number line. |a| (read the absolute value of a) 6 Absolute Value • The distance a value is from 0 is called it’s absolute value. • We write |a|, read the “absolute value of a” to represent the units a is from 0. −10 = 10 10 units –10 10 units 0 10 Use the order of operations to simplify 3 ( ) ( ) 2 ⋅ 42 − 7 4 −2 − 5 −6 ( 3 − 7 )( 4 − 6 ) 2 −5 − 4 − 5 52 − 32 2⋅6 − 4 5 8 − 3⋅ 7 ( ) Evaluate the following expressions 7 + x3 when x = 2 −x 2 +x 3 when x = −3 a 3 − 4a when a = −2 a ( a − 3) 7 Using the STO key x− 3 x when x = 5 x 2 + 0.0271x − x 3 − 4x ( ) x x−3 11 when x = −3 16 when x = −5 The Mulitplication Property of − 1 − 1⋅ a = −a Negative one times a is the opposite of a The opposite of a sum -1( a + b ) = −a + ( −b ) Distributing the negative over a sum. − ( a + b ) = −1( a + b ) = −1( a ) + ( −1)( b ) = −a + ( −b ) = −a − b 8 Rewrite the following without parentheses − ( 3x + 5) 3a + 2a − ( 4a − 7 ) − ( 6a − 4b ) 4x2 + 8 − ( 2x2 − 9) 7 y − ( 2 y + 4) 4 ( x − 3) − 5 ( 2 x − 4 ) 9