Bell Work What is the definition of Trichotomy Axiom?

Answer: For any two real numbers a and b, exactly one of the following is true: a < b a = b a > b

Lesson 40: Quotient Rule for Exponents, Distributive Property of Rational Expressions that Contain Negative Exponents

Let us review our rules and definitions for exponents: x  x  x ….. = x Product Rule If x is not zeros, then x  x = x x = x If x is not zero, then x = 1/x If x is not zero, then x = 1 m n m + n 1 -n n

The quotient rule for exponents is really an extension of the next to last definition that says if n is any real number and x is any real number that is not zero, then x = 1/x. -n n

If we wish to multiply x times 1/x We can use the definition of x to write 1/x as x and then multiply by using the product rule. x  1/x = x  x = x = x 2 5 -n 2 -2 5 2 5 -2 5 - 2 3

The quotient rule permits the same procedure in just one step The quotient rule permits the same procedure in just one step. Quotient rule for exponents: If m and n are real numbers and x ≠ 0, then x /x = x = 1/x m n m - n n - m

Example: Simplify (write the answer with x in the numerator) x 6 4

Answer: x x = x 6 -4 6 - 4 2

Practice: Simplify (write the answer with the x in the denominator): x 6 4

Answer: 1/x x = 1/x 4 -6 4 - 6 -2

Practice: Simplify x 6 -4

Answer: 2 possible outcomes 1st = x 2nd = 1/x 6+4 10 -4-6 -10

Practice: Simplify x y z z y x -5 6 -3 2

Answer: 4 possible answers: x y z 1 y z x -6 4 4 6 -4 -4 4 4 6 -6 -4

We have been using the distributive property to expand expressions that contain fractions. In this lesson, we will be doing the same thing, but not we will also consider rational expressions that contain negative exponents.

Example: Use the distributive property to expand Example: Use the distributive property to expand. Write the answer with all exponents positive. 4x y x - 3x y y -2 4 2 4 4 -2

Answer: First distribute to multiply Answer: First distribute to multiply. = 4x y x - 12x x y y y Then simplify 4 - 12x y -2 4 2 -2 4 4 4 -2 2 2

Practice: Use the distributive property to expand Practice: Use the distributive property to expand. Write the answer with all variables in the denominator. k b ab - 4p p k b 2 -1 2 -2 2

Answer: = k bab - 4p k b p k p b = 1 - 4 a p p k 2 -1 2 2 -2 2 -2 -1 -2 -4 -2

HW: Lesson 40 #1-30