Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
CC Geometry H Aim #16: How do we simplify radicals, and rationalize the denominator of a fraction? Do Now: 1. Simplify: a. b. c. d. 2. Circle the radical expressions that are irrational: A square root of a number may be negative, however the square root a number always refers to the principle square root or the positive square root of a number. i.e. -2 is a square root of 4, but THE square root of 4 is 2. Definition of the square root: If x ≥ 0, then is the non-negative number p so 2 that p = x . This definition gives us four facts: Fact 1: = a if a ≥ 0. Ex: = 12 Fact 2: = -a if a < 0. (the "opposite" of a, not negative a). Ex: Fact 3: = |a| for all real numbers a. Ex: Fact 4: = n = |13|, n = a where a is nonnegative. Ex: 1) Simplify as much as possible: 2) Compare the value of to the value of **For non-negative real numbers a and b, . = 12 = |-13| To simplify a radical means to first express the radicand as a product -- one factor a perfect square-- and then simplify. 3) Express each radical in simplest radical form: b. a. c. d. Simplifying a radical containing variables Case 1: even exponents Copy the base, ___________the exponent by ___. 4) Simplify. a. b. c. d. e. f. Case 2: odd exponents 5) Simplify. a. b. c. 3) Compare the value of to the value of **For nonnegative real numbers a and b, when b ≠ 0, . . To express a number in simplest radical form, there cannot be a radical in the denominator. If there is a radical in the denominator, we must rationalize it! **To rationalize (make rational) the denominator, it is sometimes necessary to multiply the numerator and denominator by a factor that makes the denominator a perfect square.** Example 1: Example 2: Example 3: Marcus found the scale factor of a dilation to be . When he compared his answer to Yelena's, which was , he told her that one of them must have made a mistake. Prove that their scale factors are equivalent. Example 4: Example 5: Example 6: Mixed Practice Simplify each expression as much as possible and rationalize denominators when applicable. 1. 4. 2. 3. 6. 7. 8. 10. 11. 13. 14. 9. 12. 15. 17. 19. Find, in simplest radical form, the length of the shorter leg of a right triangle if the longer leg is 25 and the hypotenuse is 30. Let's Sum it Up! To "rationalize a denominator" means that a fractional expression must have a rational number in the denominator. Name_______________________ Date: ______________ CC Geometry 10H HW #16 Express each number in simplest radical form, and rationalize denominators where applicable. 4. 5. 8. 9. Prove that are equivalent. 2 10. The area of a square is 54 cm . Express the length of one side, in simplest radical form. 11. Determine the exact area of the shaded region below: 12. Find, in simplest radical form, the length of the hypotenuse of a right triangle if the lengths of the two legs are 20 and 12 ft. Review: 1) In ΔSUN and ΔRAY, . Which pair of corresponding angles would be needed to be congruent to prove the triangles similar? 2) Find the value of x: 5x + 2 < 28 < < 8 16