Hints for solving the problems(8.2)

Negative Exponents

Until further notice don’t put an answer with a negative exponent.

      Move the negative exponent, if it is in the numerator to the denominator or if it is in the denominator to the numerator and change the sign to positive.

           Notice only the factors with the negative exponent are moved up to the numerator or down to the denominator.

Problem No. 4 - Find:      =      the ans.

Problem No. 8 - Find:      =      =   2³   =   8   the ans.

Problem No. 12 – Find:  

                                 =      the ans.

Problem No. 19- Find:    Move the factors up on down to change to positive exponents insides the parenthesis first.    Next take the reciprocal of the fraction and make the outside exponent positive.

      Now raise a power to a power by multiplying the exponents.

        the ans.

Problem No. 24 - Rewrite using only positive exponents:

      Change the negative exponents to positive exponents inside the parenthesis first.      Next invert the first fraction to get a positive exponent on the outside.      Now raise a power to a power.      Multiply the two fractions      Reduce      the ans.

Problem No. 26 - Write in scientific notation: 0.000057 Scientific notation is a way of writing very large or very small numbers in such a way that you have a one digit number (1 ≤ x < 10) times . The above problem, Count the number of positions when you move the decimal to the right and stop between the 5 and 7. That is 5. So the answer is:

  5.7 x    The exponent of 10 is negative, because the decimal was moved from left to right. If the decimal was moved from right to left, (to get one digit to the left of the decimal) the exponent of 10 would be positive.

Problem No. 28 - The following number is written in scientific notation. Write it in expanded form:  

The decimal will be moved to the right if the exponent of 10 is positive and moved to the left if the exponent of 10 is negative. In this problem the exponent of 10 is negative, so move the decimal 4 places to the left.

        The answer is:    .0001785

Multiplying and Dividing

Problem No. 30 - Reduce to lowest terms:    Notice the numerator and denominator look the same, but the signs are different. Factor out a negative one from the denominator.    Now reduce the fraction by crossing out the x - 3 in numerator and in the denominator leaving      the ans.

Problem No. 33 - Reduce to lowest terms:    Factor the numerator and factor a -1 from the denominator and factor it more.  =  -   Now reduce by crossing out a common factor in the form of a binomial (x - 7)    -    the ans. (don’t forget the negative sign)

Problem No. 38 - Find:    Factor each numerator and denominator complete and reduce by crossing out the common factors.

     Cross out the common factors: (x + 3), (x - 6) and (x - 4) The answer is:  

Problem No. 48 - Find:      To divide rational expressions, invert the second fraction and change the ÷ to ·, then factor the numerator and denominator of each fraction, cancel like terms, and multiply.

           =        the ans.

Problem No. 52 – Simplify the complex fraction below. Write your answer in lowest terms.

   To simplify a complex fraction, first rewrite it as a division problem. Then invert the divisor (the second fraction) and change the ÷ to ·. Finally, factor the numerators and denominators, cancel common factors, and simplify.      =  = = the ans.

Adding and Subtracting

Problem No. 58 - Find the LCM of    x² + 11x + 28   &   x² + 2x - 8

To find the LCM, factor each polynomial and list each factor the greatest number of times it appears in any factorization. The LCM is the product of the factors you chose.

        x² + 11x + 28   =   (x + 4)(x + 7)

        x² + 2x - 8        =   (x + 4)(x - 2)

The different factors are (x +4), (x + 7), & (x - 2) The factors are only listed once in each group, so the product of them is the LCM.

     The answer is:    (x + 4)(x + 7)(x - 2)

Problem No. 62 - Find:    

You can’t add two fractions unless they have the same denominator. The LCD of the fractions, in this case is 40mn. Multiply the first fraction (numerator and denominator) by 5n and the second fraction by 4m.

             the ans.

Problem No. 70 - Find:      

Factor each denominator first.   

The LCD is: (x + 4), (x + 3), & (x - 3). Raise each fraction so it has the LCD in the denominator.

   Multiply the numerators and combine like terms.             the ans.

Problem No. 80 - Find:      Factor each denominator and determine the LCD.      The LCD is (x - 6),

(x - 2), & (x + 3). Raise each fraction so it has the LCD in the denominator.

Now watch out, there is a negative between the fractions and it changes all the sign is the terms in the numerator of the second fraction.

           the ans.

Problem No. 83 - Simplify this complex fraction:   

To simplify a complex fraction first do the addition or subtraction in the numerator and denominator, to change them to one term. The problem can be changed to a division problem only when there is one term in the numerator and the denominator.

         =        =    Change to a division problem       Invert and multiply      Reduce      the ans.