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SIMPLIFYING CUBE ROOT FRACTIONS
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Topic 9.2 (Rational Exponents) 

     Every positiveinteger has a positive and negative square root.

     The square rootof:     16   =    4

     The square rootsign:    is called the root symbol.

     16 is calledthe radicand

     The number inthe v of the radical is called the index number or root number. If

      there isnt any number show it is 2,  otherwise if you have a cube root& etc.

       the3 will be written in the index (root) number position.

A square root of a negative numbercan never be a real number.

      REMEMBER:   The square root of 16 is 4,  why , because 4 squaredis 16.

       Therefore: the square root of -16 cannot be found in the real number system,

          because 4 or -4 squared  will not be 16.

            25    =    5      The primary or principal root which is alwayspositive.

       -  25    =   -5       The secondary root which is alwaysnegative.

        25    =   5      The double root which is always positive andnegative.

            If the index (root) number is even, a negative number will not result in a real

                number.

            If the index (root)  number is odd, a negative number will work, giving a

                negative answer.      - 8    =     - 2

             The above reads the cube root of   -8   is   -2.

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              am/n   =   (    a)m  =   (am)This says the nth root of a to themth power. 

                      The nth root of a.

               82/3   =   (     8)2  =       (82)  This says the cube root of 8 squared.

                    The cube root of 8.

               When solving this kind of problem change the original to its corresponding

                   radical.

      REMEMBER:The numerator of the fraction is the exponent of the radicand and

                               the denominator is the index (root) number.

                             It is easier to change the fractional exponent to a root raised to a

                               power. This means is easier to take the root and than raise it to

                               the power.

                                82/3   =     (      8)2 = (2)2 = 4

                                     The cube root of 8  =  2

                                      165/4 = (     16)5 = (2)5 =32

                                    The fourth root of 16 = 2.

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NEGATIVE EXPONENTS

      x-n  =    1/xn    and 1/x-n =  xn      This means when the exponentis negative in

                                                                  the numerator (upstairs), allterms affected by

                                                                  the negative exponents are moved to the

                                                                  denominator (downstairs) and the exponents

                                                                  are changed to a positive exponent

                                                                  Also when the terms have a negative 

                                                                  exponent in the denominator they are moved

                                                                  to the numerator and the exponent is 

                                                                  changed to a positive exponent.

Example:  (All answers must have apositive exponent.)

 

   2x-2y3/ w-1z-3               Move the x term to the denominator, and the w term to

                                           the numerator and change the sign of the exponents.

    2wy3 / x2z3                   Ans.

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   (2-1x4y-3/  z-2)-2            Move the terms with the negative exponents up or down

  (x4z2 / 2y3)-2                   and change the exponent to positive. (terms inside the

                                            parenthesis.

(2y3 / x4z2)2                      Next turn the whole fraction upside down and change the

                                            sign of the outside exponent to positive.

                                            Now raise a power to a power.

  4y6 / x8z4                         Ans.

There is also a way to work theseproblems by raising a power to a power right at 

the start and them changing to apositive exponent.  The only problem with this

is you might miss writing down anegative at some point and the problem will be

wrong.

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SIMPLIFYING RADICALS

To simplify a radical expression

Make sure all perfect roots are taken out.

No fractions are left under the radical sign.

The denominator of a fraction doesnt contain a radical.

The index number and the exponent of the radicand do not

  contain a common factor.

    (72)     =    (36 2)     =      62

                       (8x4y12z5)    =    (4 2x4y12z4z)  =     2x2y6z2 (2z)

                       (242x7y3z15)  =     (121 2x6xy2yz14z)  =      11x3yz7 (2xyz)

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(1/5)   change to   1/5 ,   multiply the numerator and the

       denominator by   5.

           5/  (25)     =       5 /  5.

 (3/2)   =   3 / 2 multiply the numerator and the denominator by   2.

           6 / 4    =     6 /   2.

A short cut: Multiply the numerator by the denominator and put this in

     in the numerator of the answer under the radical and put the old

     denominator in the denominator of the answer without the radical. 

      (3/5)    =      (15) /  5

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 5/ 7)   multiply the numerator and the denominator by   7.

                    5 7 /  49    =    5 7/  7

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          (6/ 5)   multiply the numerator and the denominator by   5.

           (30)/  (25)     =       (30) /  5.

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   (32)      =    3        divide the exponent by the index no. 2

            (74)     =    72     divide the exponent by the index no. 2.

_________________________________________________________________________

ADDING AND SUBTRACTING RADICALS

To add or subtract radicals you musthave the same radicand and the

      sameindex number.

    22   +   52   -   32  =   42.

    8    +    (18)   -     (50)       Simplify allterms (remove all perfect

                                                            squares)

        (4x 2)  +    (9x 2)   -    (25x 2)

            22    +         32    -        52  =   0

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 CONJUGATES

In a problem like:    5 /(2  + 7) You must multiply the numerator and the

     denominator by 2  - 7.

This is called the conjugate. Theconjugate is the same two numbers with the sign

      betweenthem changed.

      Suchas:     5   -   3 theconjugate is   5    +    3.

                           2   +    5 the conjugate is    2    -    5.

   So in the problemabove           5  (2-  7            =        10    -    57       =

                                                    (2   +  7)(2   -  7)                 4   -   7

                                                          10   -   57      =           -10  +   57

                                                                  -3                                        3