Complete the following questions to practice finding non-permissible values.

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  1. Simplify each of the following rational expressions. Remember to state the non-permissible values.


    1. «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mrow»«mn»3«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mi»y«/mi»«/mrow»«mrow»«mn»15«/mn»«mi»x«/mi»«mi»z«/mi»«/mrow»«/mfrac»«/math»
      «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable rowspacing=¨20px 20px 20px¨ columnalign=¨center left¨»«mtr»«mtd/»«mtd»«mfrac»«mrow»«mn»3«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mi»y«/mi»«/mrow»«mrow»«mn»15«/mn»«mi»x«/mi»«mi»z«/mi»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mn»3«/mn»«mi»x«/mi»«mfenced»«mrow»«mi»x«/mi»«mi»y«/mi»«/mrow»«/mfenced»«/mrow»«mrow»«mn»3«/mn»«mi»x«/mi»«mfenced»«mrow»«mn»5«/mn»«mi»z«/mi»«/mrow»«/mfenced»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«menclose notation=¨updiagonalstrike¨»«mn»3«/mn»«mi»x«/mi»«/menclose»«mfenced»«mrow»«mi»x«/mi»«mi»y«/mi»«/mrow»«/mfenced»«/mrow»«mrow»«menclose notation=¨updiagonalstrike¨»«mn»3«/mn»«mi»x«/mi»«/menclose»«mfenced»«mrow»«mn»5«/mn»«mi»z«/mi»«/mrow»«/mfenced»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mi»x«/mi»«mi»y«/mi»«/mrow»«mrow»«mn»5«/mn»«mi»z«/mi»«/mrow»«/mfrac»«mo»,«/mo»«mo»§#160;«/mo»«mi»x«/mi»«mo»§#8800;«/mo»«mn»0«/mn»«mo»,«/mo»«mo»§#160;«/mo»«mi»z«/mi»«mo»§#8800;«/mo»«mn»0«/mn»«/mtd»«/mtr»«/mtable»«/math»


    2. «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mrow»«mn»1«/mn»«mo»-«/mo»«mi»a«/mi»«/mrow»«mrow»«mn»20«/mn»«mo»-«/mo»«mn»20«/mn»«mi»a«/mi»«/mrow»«/mfrac»«/math»
      «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable rowspacing=¨20px 20px 20px¨ columnalign=¨center left¨»«mtr»«mtd/»«mtd»«mfrac»«mrow»«mn»1«/mn»«mo»-«/mo»«mi»a«/mi»«/mrow»«mrow»«mn»20«/mn»«mo»-«/mo»«mn»20«/mn»«mi»a«/mi»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mn»1«/mn»«mo»-«/mo»«mi»a«/mi»«/mrow»«mrow»«mn»20«/mn»«mfenced»«mrow»«mn»1«/mn»«mo»-«/mo»«mi»a«/mi»«/mrow»«/mfenced»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mmultiscripts»«menclose notation=¨updiagonalstrike¨»«mn»1«/mn»«mo»-«/mo»«mi»a«/mi»«/menclose»«mprescripts/»«none/»«mn»1«/mn»«/mmultiscripts»«mrow»«mn»20«/mn»«mfenced»«menclose notation=¨updiagonalstrike¨»«mn»1«/mn»«mo»-«/mo»«mi»a«/mi»«/menclose»«/mfenced»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mn»1«/mn»«mn»20«/mn»«/mfrac»«mo»,«/mo»«mo»§#160;«/mo»«mi»a«/mi»«mo»§#8800;«/mo»«mn»1«/mn»«/mtd»«/mtr»«/mtable»«/math»


  2. Explain why you must show the non-permissible values as part of the solution when simplifying a rational expression.
    The non-permissible values are the numbers that will make the rational expression have a denominator of zero. If the denominator is equal to zero, then the expression is undefined because dividing by zero is undefined. The simplified expression is valid only when it does not have a denominator equal to zero.


  3. Why must you find the non-permissible values before you simplify the rational expression?
    To find the non-permissible values before you simplify is important because you are cancelling factors when you simplify. This cancelling may eliminate some of the factors that produce the non-permissible values. If you find the non-permissible values after simplifying, you will have “lost” some of the non-permissible values. For example, in 2(a) shown on the previous page, you would not have «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»x«/mi»«mo»§#8800;«/mo»«mn»0«/mn»«/math» if you found the non-permissible values after simplifying.