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

2. 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.

3. 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!

4. 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

5. 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

6. 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

7. 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

8. Properties of Radicals
Product Rule for Radicals

9. Properties of Radicals
Quotient Rule for Radicals

10. Simplifying Radicals Factor the radicand
Group these factors in sets numbering the same as the index
Use the Product Rule or Quotient Rule for Radicals to rewrite the expression
Simplify (when the index and the radicands exponent match, the radical simplifies as an exponentless radicand)

11. Rationalizing Denominators
Simplifying Radicals: A radical is considered simplified when:
The radical contains NO fractions and NO negative numbers
NO radicals appear in the denominator of a fraction
The technique we use to get rid of any radicals in the denominator of a fraction is called rationalizing.
To rationalizing denominators, we are going to multiply the denominator by something so that the index and the radicands exponent match meaning the radical(s) in denominator will simplify as an exponentless radicand.
Remember … if you multiply the denominator times something, you must multiply the numerator times the exact same thing!

12. CONJUGATES (rationalizing denominators)
These first four examples had only ONE term in the denominator. If there are two terms, there is a slightly different technique required in order to rationalize the denominators.
We are going to multiply the denominator by its CONJUGATE (Remember … if you multiply the denominator times something, you must multiply the numerator times the exact same thing!)