Pre-Calc 11: Week 14 Multiplying and Dividing Rational Expressions

Last week in Pre-Calculus 11 we learned about multiplying and dividing rational expressions. A thing to remember about expressions is that it does not have an equal sign and you do not need to solve, you only need to simplify. I will be teaching you what non-permissible values are and how to determine what they are.

Non-permissible value :  Values that cause the fraction to have a denominator with a value of zero. (In math, we cannot divide by zero).

Here is an example of multiplying rational expressions


Step 1:  Simplify \frac{3x}{2(x-3)}\cdot\frac{8(x-3)}{9x^2}

Non permissible values: x\neq -3

  • \frac{3x}{2(x-3)}\cdot\frac{8(x-3)}{9x^2}
  • remove (x-3) from top and bottom because they cancel eachother out
  • then it becomes… \frac {3x}{2}\cdot\frac{8}{9x^2}


Step 2: Simplify by multiplying across (Just do it)

  • \frac {3x}{2}\cdot\frac{8}{9x^2}
  • \frac{24x}{18x^2}

Step 3: Take the highest common factor from both the numerator and the denominator.

In this case 6 is the highest number that goes into both.

  • \frac{24x}{18x^2}
  • \frac{4x}{18x^2}

Step 4: Notice that x is on the bottom and the top, if it has a pair it can cancel out. two x’s on the bottom one on the top. when they cancel out each other you will be left with only 1 on the bottom.

  • \frac{4x}{3x^2}
  • \frac{4}{3x}

Final Answer: \frac{4}{3x}

Dividing Rational Expressions 

There are many steps when dividing rational expressions

  1. Simplify the fraction: can factor or take out the common denominator.
  2. State the non permissible values.
  3. Reciprocate the second fraction and it will because a multiplication expression.
  4. State the restrictions again because there are new values in the denominator and could be non-permissible.
  5. Simplify (cancel out like terms that have a pair on the numerator and denominator.
  6. Multiply across (Just do it)
  7. Simplify again if possible.

Step 1: Simplify \frac{x+5}{x-4}\div\frac{x^2 - 25}{3x-12}


  • \frac{x+5}{x-4}\div\frac{x^2 - 25}{3x-12}
  • \frac{x+5}{x-4}\div\frac{(x+5)(x-5)}{3(x-4)}

Step 2: Non-permissible values

  • x\neq 4

Step 3: Reciprocate

  • \frac{x+5}{x-4}\div\frac{(x+5)(x-5)}{3(x-4)}
  • \frac{x+5}{x-4}\cdot\frac{3(x-4)}{(x+5)(x-5)}

Step 4: Non-permissible values

  • x\neq 4
  • x\neq 5
  • x\neq -5

Step 5: Cross out like terms

  • \frac{x+5}{x-4}\cdot\frac{3(x-4)}{(x+5)(x-5)}
  • \frac{3}{(x-5)}

Step 6: Multiply across if possible

  • In this example it is not

Step 7: Simplify further if possible

  • In this example it is not

Final Answer:  \frac{3}{(x-5)}


This is how you Multiply and Divide Rational Expressions



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