Lecture 1.4 to 1.5 bt
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Today's agenda covers rational expressions, exponents, and radicals. It includes attending to announcements about an upcoming exam, working through examples of rational expressions, exponents, and radicals, as well as assigned homework from the MyLabsPlus website due on Sunday. The class will work through examples of factoring polynomials, rational expressions, exponents, and radicals.
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This document discusses key concepts in geometry related to ratios, proportions, and similarity. It introduces ratios and proportions, covering verifying proportions using cross multiplication and solving proportions. It then discusses similar shapes, geometric means, and the extreme and mean ratio property. Several constructions are presented for the extreme and mean ratio, including using squares, triangles, and the golden ratio. Homework problems are assigned.
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This document discusses geometry topics including proving triangles similar using AAA, SAS, and SSS similarity postulates, volume formulas for conic frustums, properties of inscribed right triangles, mean proportions, indirect measurement techniques, and the midsegment and side splitter theorems. It also announces an upcoming exam and homework assignments covering sections 6.2 and 6.3 that are due on Tuesday.
Geo 6.1 bt
This document discusses various topics in modern geometry including:
- Ratios and proportions, and methods for verifying and solving proportions.
- Geometric mean and finding the geometric mean of two numbers.
- The extreme and mean ratio, and constructions for drawing a point B given the ratio of AB to AC.
- Fibonacci numbers and the ratio of successive numbers.
- The golden ratio φ and its relationship to dividing a line segment in the golden ratio.
- Applications of the golden ratio including the golden rectangle and golden spiral.
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Lecture 1.4 to 1.5 bt
- 1. Today’s Agenda • Attendance / Announcements – Chapter 1 Exam on Monday 1/26 • Questions from HW? • Section 1.4 Rational Expressions • Section 1.5 Exponents
- 4. Factoring Polynomials Special Cases Difference of Squares (p. 24) 𝑥2−𝑦2 = 𝑥 + 𝑦 𝑥 − 𝑦 𝑥2 − 25
- 5. Factoring Polynomials Special Cases Difference of Squares 𝑥2 − 𝑦2 = 𝑥 + 𝑦 𝑥 − 𝑦 9𝑝2 − 16
- 6. Factoring Polynomials Special Cases Difference of Squares 𝑥2 − 𝑦2 = 𝑥 + 𝑦 𝑥 − 𝑦 49𝑎2 + 9
- 7. Factoring Remember to always factor out the G.C.F. and keep factoring until you can’t factor anymore! xxx 963 23
- 8. 1.4 Rational Expressions An expression that can be written as the quotient of two polynomials. 2 532 2 x xx 1 8 x x 2 note*: We need to make sure…..
- 9. Simplifying Rational Expressions Cancellation Property Q P QS PS
- 12. Simplify each expression… 9𝑚 27𝑚3 10𝑧+5 20𝑧+10 𝑟2−𝑟−6 𝑟2+𝑟−12 3 27 9 m m 2 1 3 1 m 2 3 1 m 1210 125 z z 2 1 34 23 rr rr 4 2 r r
- 13. Simplify each expression… 𝑧2 + 4𝑧 + 4 𝑧2 − 4
- 14. Multiplying Rational Expressions (p. 29) Same as fraction arithmetic “Multiply straight across…top and bottom”
- 16. Dividing Rational Expressions (p. 29) Same as fraction arithmetic “Dividing by a fraction Multiply by the reciprocal”
- 17. Perform each operation… 6𝑥2 𝑦 2𝑥 ÷ 21𝑥𝑦 𝑦 𝑛2 − 𝑛 − 6 𝑛2 − 2𝑛 − 8 ÷ 𝑛2 − 9 𝑛2 + 7𝑛 + 12
- 18. Adding / Subtracting Rational Expressions “Just like with fractions…We need a….”
- 20. Complex Fractions (p. 33) 1 3 2 x x
- 21. Section 1.5 Exponents More Properties of Integer Exponents
- 23. Exponents…continued More Properties of Integer Exponents
- 25. Exponents…continued Negative Exponents We usually want our answers to be written only in Positive Exponents…So we need to rewrite.
- 29. Properties of Exponents • You’ll need to know the entire table of properties on page 41.
- 30. Rational Exponents Radicals 𝒂 𝟏 𝒏 is defined to be the “nth root of a.” You can think of fraction exponents as radicals…and vice versa
- 31. Rational Exponents Radicals x “Invisible Numbers!”
- 32. Rational Exponents (p. 44) So, if n is an even integer and a ≥ 0; or if n is odd, then n aan 1 When dealing with radicals/roots, we always need to think about…
- 33. Rational Exponents Radicals And if the root exists… n m aan m
- 34. Rewrite in radical notation 3 2 x 3 1 5x 3 2 x 3 5x
- 35. Rewrite in exponent notation 5 x 3 7 3 3 x
- 36. Properties of Radicals (p. 44) Radicals can be separated across multiplication and division.
- 37. Properties of Radicals We use these properties when we simplify roots. 72
- 38. Properties of Radicals We use these properties to simplify roots. 155
- 39. Properties of Radicals 80245205 Remember, we can only add “like” terms
- 40. Properties of Radicals 2353
- 41. Classwork / Homework • Classwork –Page 71: 51 – 93 odd • Homework – MyLabsPlus HW1 »Due Sunday 1/18, 11:59pm • Read 1.6, 1.7 (No Class Monday 1/19) • Quiz Today (extra credit)