Algebraic Fractions: Unit 9 Study Sheet Rational Expressions and Types of Equations Simplifying Algebraic Fractions: To simplify an algebraic fraction means to reduce it to lowest terms. This is done by dividing out the common factors in the numerator and the denominator. {This means that you need to factor both the numerator and denominator first.} o Common factors divide to one x 1 o Opposite factors divide to negative one: 1 1 x Multiplying Algebraic Fractions: Factor each numerator and denominator. The rule for multiplying algebraic fractions is the same as the rule for multiplying numerical fractions multiply the tops (numerators) AND multiply the bottoms (denominators). You can only cancel top with bottom or bottom with top. There is NO canceling bottom with bottom or top with top.
Dividing Algebraic Fractions: The rule for dividing algebraic fractions is the same as the rule for dividing numerical fractions Change the division sign to multiplication, take the reciprocal of the second fraction ONLY, and then follow the steps for multiplying algebraic fractions. Adding/Subtracting Algebraic Fractions: The basic rule for adding and subtracting fractions is to get a common denominator first {the smallest number that both denominators can divide into without remainders}. Once you get a common denominator, remember to just add/subtract numerators {keep the common denominator the same}. You might need to factor the denominators to help get the LCD. When rewriting fractions so that they have common denominators, remember whatever is multiplied times the bottom must ALSO be multiplied times the top. {Multiplying the top and bottom by the same number is multiplying by the multiplicative identity element (which is = 1) and therefore does not change the value of the fraction.}
Undefined Fractions: A fraction is undefined when the denominator equals zero. Solving Fractional Equations: Method 1: When a single fraction is equivalent to a single fraction Cross multiply and solve. Remember to check for extraneous roots by substituting each answer into every denominator to check for values that create undefined fractions. Method 2: When a sum or difference of fractions are involved Find the LCD for each fraction involved in the equation Once you find the LCD, multiply each fraction by the LCD to clear the denominator Solve the resulting equation Remember to check for extraneous roots by substituting each answer into every denominator to check for values that create undefined fractions.
Word Problems Involving Rational Equations Problems Involving People or Machines Doing Work When problems involve two people working together on a job then their rates add and they can perform the job working together in a shorter amount of time. If we let x = time it takes 1 person to complete the task, then his work rate is. In other words, he can complete the 1 job in x number of hours.
Problems Involving Percentages When problems involve percentages, think about setting up a proportion.
Problems Involving Moving Objects When objects are in motion, a variation of the distance formula must be used. This formula can be manipulated in order to change it to a formula that will give a rational expression for the time. This variation is Distance divided by Rate equals Time.
An exponential expression is one which contains an exponent. Fractional exponents: b a a b x x missing a means a = 2 missing b means b = 1 To solve equations with fractional exponents: Isolate the variable with the fractional exponent. To eliminate the fractional exponent, raise each side to the reciprocal power {remember not to change the sign}. Remember: If the denominator of the reciprocal power is an even number, you include the answer in your final solution. Solve the resulting equation. Check your answer(s) to avoid extraneous roots.
To solve exponential equations with the same base: Example 1: Example 2:
Solving Radical Equations: