7.1 nth Roots and Rational Exponents by zqa20601

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									                               7.1 nth Roots and Rational Exponents

     na
                  3
                      8
Roots:
Powers:



Real nth roots:
For all   n= integer >1        a = real number
n = odd: “a” has ONE real root          n = even & a > 0:                         ‘a’ has 2 real roots
      3                                                           4
          8                                                           16
                                                           n = even & a = 0:     ‘a’ has one root
                                                                  4
                                                                      0
                                                           n = even & a < 0:     ‘a’ has no real roots
                                                                  6
                                                                      −64

Finding the real nth roots:
1) nth root on calc:                                       2) raise it to the (1/n) power



Ex. 1: The 5th real root of –32                            Ex.2 : The 3rd real root of 64



The exponent doesn’t have to be in the form of (1/n) to be rational.
Ex. 3: The 5th root of 325               Ex. 4: The 3rd root of 46



Rational Exponents:
          1
Let a         n
                      be an nth root of ‘a’, and let ‘m’ be a positive integer
              1                   1                m
n
    a =a          n
                           → (a n ) m         →a       n



          m                1
So: a             n
                      = (a n )m = ( n a ) m
Evaluate:
Ex. 5:                             Ex. 6:                            Ex.7:
     5                                  2                                  −4
         2                                  3                                   3
16                                 64                                216




Solving Equations using nth roots:
Solve normally to get the variable with the root alone
Raise both sides to the reciprocal power
Remember ± if needed

Solve:
Ex. 8:                             Ex. 9:                            Ex. 10:
     4                                          3
6 x = 3750                         ( x + 1) = 18                     5 y 4 = 80




Ex. 11: A basketball has a volume of about 455.6 in3 . The formula for the volume
of a basketball is V = 4.18879 r 3 . Find the radius of the basketball.
                          7.2   Properties of Rational Exponents
Let the bases ( a & b) be real numbers and the exponents (m & n) be rational
numbers.
                                                         1                   5
             aman =                              Ex: m       2
                                                                 *m              2
                                                                                     =

                                                                 1
             (a m ) n =                          Ex: (c 3 )          3



                                                                             3       4
             (ab) m =                            Ex: (27 x 3 yz 8 )                      3




                                                                 −1
                 a −m =                          Ex:     36              2



                                                                 5
             am                                              x       3
                =                                Ex:             2
                                                                             =
             an                                              x       3




                                                                         1
                    m
             a                                      8m  3
               =                               Ex:      =
             b                                      n 




Simplify:
Ex.1:                                            Ex.2:

 1       1                                                    −1
6 2 *6       3
                                                 (43 * 23 )        3




Understanding the properties of radicals:
                                     3                                                       5
 4                                       82                                                      210
Simplest form: keep going until you can’t go any further
      Get rid of negative exponents, reduce, rationalize denominators, remove any
      nth powers

Need to know your perfect squares, cubes, 4th powers, 5th powers, etc




Ex.3:                                           Ex.4:                              Ex.5:
                                                3
                                                     32
(   3
        25   )( 5 )
               3
                                                 3
                                                      4
                                                                                   4
                                                                                       64




               5
Ex. 6:             1215                         Ex. 7:                             Ex.8:
                                                                                       7
                                                3 4 24 *5 4 2                      4
                                                                                       8




Combining Roots and Radicals- treat them as like terms
        Ex. 9:             ( ) − 3( 4 )
                          5 4
                                3
                                    4
                                        3
                                            4
                                                          Ex. 10:   3
                                                                        81 − 3 3
Simplifying Expressions using Variables
A variable could be positive, negative or zero
Assume all variables are positive
Ex. 11:                             Ex. 12:                                    Ex. 13:
                                                                                       2

3                                         x5                                   18rs 3
    27 z 9                            5
                                                                                   1
                                          y10
                                                                               6 r 4 t −3




Ex. 14:                     Ex. 15:                                  Ex. 16:
                                  2
                                 g
4
    12d 4 e9 f 14            5                                       8 x −3 x
                                 h7




Ex. 17:                                         Ex. 18:
       1        1
       4
3 gh − 6 gh     4
                                                2 4 6 x5 + x 4 6 x
                 7.3  Power Functions and Function Operations
Power Function- form of y = ax b
 ‘a’ is a real number
 ‘b’ is a rational number

Function Operations- adding, subtracting, multiplying, dividing functions

Ex. 1:
                1                     1
Let f ( x) = 3 x 3 and g ( x) = 2 x 3
Find a) the sum                                   b) the difference




Ex. 2:
                 1                1
Let f ( x) = 4 x and g ( x) = x
                 3                2

Find a) the product                               b) the quotient




Composition of two functions f(g(x))….read :
One function following another
Ex.3: Let f ( x) = 3 x −1 and g ( x) = 2 x − 1
         Find: A) f(g(x))                                B)   g(f(x))




Ex. 4: Let f ( x) = 2 x −1 and g ( x) = x 2 − 1
         Find: A) g(f(x))                               B) f(g(x))
                            7.4       Inverse Functions
Recall: function-mapping such that each x-value is paired with exactly one y-value.
         Vertical line test

INVERSE RELATION- each y-value is paired with the original x-values.
     The domains and ranges are reversed

Notation:

The graph of an inverse relation is the reflection of the original over the line y = x.

Ex. 1:
a) Graph: y = -3x +6
b) switch the x and y values around and then graph the new line
c) graph the line y = x.




Finding the inverse equation with respect to x and y.
       Switch x and y and then solve for y if possible



Ex. 2:                                                     Ex. 3:
Find the inverse of y = -3x +6                             Find the inverse of
                                                              1
                                                           y = x −1
                                                              3




Determining if two functions are inverses of each other.
1. Two functions ‘f’ and ‘g’ are inverses iff f(g(x))= x and g(f(x)) = x
2.
3.

                                            1
Ex. 4: Verify that f(x) = 3x+3 and g(x) =     x − 1 are inverses.
                                            3
The inverse of every linear function is also a function.
The inverse of every quadratic function is NOT always a function



Ex. 5: Find the inverse of f(x) = x 4 and then graph both.




Ex. 6: Find the inverse of f(x) =   x 4 , x ≥ 0 and graph both.




Ex. 7: Determine whether the inverse of f ( x) = 2 x 2 − 4 is a function and then find
the inverse.
                               7.6            Solving Radical Equations
Solving radical equations-
   1) isolate the radical as much as possible
   2) either
           a. raise each side to the power of the root
           b. turn the root into a power &then raise each side to the reciprocal pwr
   3) Check solutions-some times you get EXTRANEOUS SOLUTIONS: you
       followed all the operations correctly, but it doesn’t work when you check it.
       If none of the solutions work, then your answer is NO SOLUTION.

Ex. 1:                           Ex. 2:                            Ex. 3:
                                          4
  5− 4 x = 0                         3x       3
                                                  = 243              2x + 8 − 4 = 6




Ex. 4:                                                             Ex. 5:
  4 x + 28 − 3 2 x = 0                                              x + 2 = 2 x + 28




                         7.7         Statistics and Statistical Graphs
Statistics- numerical values used to summarize and compare sets of data

Measures of Central Tendency- usually show where the center of the data is
      Mean-
            Notation:
      Median-



         Mode-
Ex.1: The number of games won by teams in the Eastern Conference for the 1997-
1998 regular season of the NHL is shown below. Find the mean, median, and mode.
36, 39, 40, 34, 48, 33, 25, 30, 37, 17, 42,40, 24




Measures of Dispersion- usually show how spread out the data is
      Range-
      Standard Deviation- describes the typical difference (deviation) between
                          the mean and an actual data value
             Notation:



Ex. 2: Find the range and standard deviation of the number of wins from the data
in ex.1: 36, 39, 40, 34, 48, 33, 25, 30, 37, 17, 42,40, 24




Ex. 3: Find the mean, median, mode, range, and standard deviation from the
following test scores: 92, 94, 87, 76, 69, 82, 62, 90, 76, 82, 85, 87, 64, 61, 95, 87
________________________________________________________
Statistical Graphs- more useful for visual people
Box and Whisker Plot-
       Box- encloses the middle half of the data
       Whisker-extends to the minimum/maximum data points

    1. Q2-Median of all data points
    2. Q1 (lower quartile)-median of lower half of the data values
    3. Q3 (upper quartile)-median of upper half of the data values
    4. Maximum/Minimum
Ex. 5: Draw a box/whisker plot of the data from ex. 1: 36, 39, 40, 34, 48, 33, 25,
30, 37, 17, 42,40, 24




Frequency Distribution- tally system that helps organize data which then can be
made into a bar graph or other types of graphs
Need a freq distribution before you can make a bar graph/histogram



Ex. 6: Draw a frequency distribution table from the data in ex.1: 36, 39, 40, 34,
48, 33, 25, 30, 37, 17, 42,40, 24




Histogram- type of bar graph that groups data into equal width-intervals
Ex. 7: Draw a histogram from the data in ex. 1: 36, 39, 40, 34, 48, 33, 25, 30, 37,
17, 42,40, 24

								
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