Exercises


  1. Fill in the table.
    Denominators Possible Common Denominator Restrictions
    • \(x^4y^8z^3\)
    • \(x^{12}y^6z^5\)
    • \(x^9y^{12}z^3\)
    		    
    • \(5x^4y^8\)
    • \(15x^2y^6\)
    • \(20x^9y\)
    		    
    • \((x+2)^3(2x+1)^4\)
    • \( (x-2)^2(x-1)^3\)
    • \( (x+2)^5(x-1)^7(2x+1)^6\)
    		    
    • \(x^2-4\)
    • \( (x-1)(x+2)^2\)
    • \( x^2-3x+2\)
    		    
    • \(x^2-x-2\)
    • \( x^2+x-6\)
    • \( x^2+4x+3\)
    		    
  2. Simplify and state restrictions.
    1. \(\dfrac{5}{a+8}+\dfrac{7}{2a}\)
    2. \(\dfrac{-3}{x+1}-\dfrac{2}{3x}+\dfrac{2}{9}\)
    3. \(\dfrac{n-2}{n+4}+\dfrac{4}{n-1}\)
    4. \(\dfrac{4x-5}{x^2-5x-24}-\dfrac{3x+1}{x^2+11x+24}\)
    5. \(\dfrac{3x-5}{6x^2+13x+6}-\dfrac{3}{3x^2+5x+2}\)
    6. \(\dfrac{1}{x-2}+\dfrac{x}{x+2}-\dfrac{2+x}{x^2+x-6}\)
  3. Simplify and state restrictions.

    \(\dfrac{\frac{1}{x^2}-\frac{1}{y^2}}{\frac{1}{x}+\frac{1}{y}}\)

  4. A pawnshop owner bought a used watch for \($\dfrac{2a}{a-2}\) and sold it for \($\dfrac{16}{a-1}\).  Determine a simplified expression to represent the amount of money that he made.
  5. Mandy scored \(11\) points on the first test and \(a^2\) points on the second test, for some integer \(a\)..  The first test was out of \(121-a^2\) and the second was out of \(a+11\).  
    1. Write an expression for her first score as a rational expression.
    2. Write an expression for her second test score as a rational expression.
    3. Write a simplified expression to represent her average for the two tests.
  6. The mean of three positive numbers can be represented by the expression \(\dfrac{43x+92}{60x(x+4)}\).  If two of the numbers can be expressed as \(\dfrac{1}{x+4}\)and \(\dfrac{2}{5x}\), determine a simplified expression for the third number.
  7. The lens formula \(\dfrac{1}{a}+\dfrac{1}{b}=\dfrac{1}{f}\) describes the relationship between the location of an object, \(a\), the location of the image, \(b\), and the focal length of the lens, \(f\).  If \(a=x+7\) and \(b=(x+7)(x-1)\), determine a simplified expression for \(f\).
  8. The total number of glass balls, \(B\), in a square pyramid whose base measures \(m\) by \(m\) glass balls is given by the formula \(B=\dfrac{m^3}{3}+\dfrac{m^2}{2}+\dfrac{m}{6}\).  Determine a simplified expression to represent the number of glass balls, \(B\), if \(m=\dfrac{1}{1-x}\).

    Balls stacked into a triangular pyramid.

    Source: Stacked Balls - porcorex/iStock/Getty Images