Unit 7 Rationals and Radicals Rational Expressions –Reducing/Simplification –Arithmetic (multiplication and division) Radicals –Simplifying –Arithmetic (multiplication, division, addition, subtraction) –Rationalizing denominators

Rational Expressions Definition:  Fractions that contain integers in their numerator and/or denominator are called rational numbers (this is a reminder from Unit 1)  Fractions that contain polynomials in their numerator and/or denominator are called rational expressions.

Rational Expressions  A rational expression's denominator can never be zero.  A rational expression's value is zero when its numerator is zero, and the only way a rational expression's value can be zero is for its numerator to be zero.  When a rational expression's denominator is 1, the value of the rational expression is the value of its numerator.  A rational expression's value is 1 when its numerator and denominator have the same (nonzero) value.  Even if a rational expression's numerator is zero, the first point applies: a rational expression's denominator can never be zero. A thing that is written ''0/0'' isn't a number. In particular, we aren't going to call it 1!  When given as a final answer, a rational expression must be reduced to lowest terms!

Simplifying Rational Expressions  Factor the numerator and denominator completely using the same factoring strategy we used in Unit 6  Factor out the GCF FIRST (section 5.5)  Count the number of terms  2 Terms: (section 5.7)  3 Terms: (section 5.6)  4 Terms: (section 5.5)  Cancel out factors that are common to both the numerator and denominator

Multiplying Rational Expressions Our Plan of Attack  Factor all the numerators and denominators  Cancel out factors common to the numerators and denominators  Multiply the numerators  Multiply the denominators

Dividing Rational Expressions Our Plan of Attack: Dividing rational expressions is very much like multiplying rational expressions with one extra step –KEEP – SWITCH - FLIP: Keep the first fraction, Switch to multiplication, Flip the second fraction upside down –Factor all the numerators and denominators –Cancel out factors common to the numerators and denominators –Multiply the numerators –Multiply the denominators

Simplifying Radicals: Based on what we saw in the nth root examples, we can see one of the keys in simplifying radicals is to match the index and the radicand’s exponent. A radical is considered simplified when: Each factor in the radicand is to a power less than the index of the radical The radical contains NO fractions and NO negative numbers NO radicals appear in the denominator of a fraction Product Rule for Radicals

Quotient Rule for Radicals

Simplifying Radicals  Factor the radicand  Use the Product Rule or Quotient Rule for Radicals to rewrite the expression  Simplify

To Multiply Radicals Must have same index Multiply the Coefficients Multiply the Radicands To Divide Radicals Must have same index Divide the Coefficients Divide the Radicands To Add or Subtract Radicals Must have Same Index and Same Radicand Add/Subtract coefficients Arithmetic of Radicals