MTH55_Lec-40_sec_7-2b_Rational_Exponents.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer Chabot Mathematics §7.2 Rational Exponents

MTH55_Lec-40_sec_7-2b_Rational_Exponents.ppt 2 Bruce Mayer, PE Chabot College Mathematics Review §  Any QUESTIONS About §7.2 → Radical Functions  Any QUESTIONS About HomeWork §7.2 → HW MTH 55

MTH55_Lec-40_sec_7-2b_Rational_Exponents.ppt 3 Bruce Mayer, PE Chabot College Mathematics Laws of Exponents  For any real number a, any real number b > 0, and any rational exponents m & n In multiplying, we can add exponents if the bases are the same. In dividing, we can subtract exponents if the bases are the same. To raise a power to a power, we can multiply the exponents. To raise a product to a power, we can raise each factor to the power. To raise a quotient to a power, raise both the numerator & denominator to the power.

MTH55_Lec-40_sec_7-2b_Rational_Exponents.ppt 4 Bruce Mayer, PE Chabot College Mathematics Example  Laws of Exponents  Use the rules of exponents to simplify. Write the answer with only positive exponents  SOLUTION Use the quotient for exponents. (Subtract the exponents.) Rewrite the subtraction as addition. Add the exponents.

MTH55_Lec-40_sec_7-2b_Rational_Exponents.ppt 5 Bruce Mayer, PE Chabot College Mathematics Example  Laws of Exponents  Use the Laws of Exponents to Simplify  SOLUTION

MTH55_Lec-40_sec_7-2b_Rational_Exponents.ppt 6 Bruce Mayer, PE Chabot College Mathematics Example  Laws of Exponents  Use the Laws of Exponents to Simplify  SOLUTION

MTH55_Lec-40_sec_7-2b_Rational_Exponents.ppt 7 Bruce Mayer, PE Chabot College Mathematics Example  Laws of Exponents  Write with only positive exponents. Assume that all variables are ≥ 0 Power rule m 1/4 n –6 m –8 n 2/3 –3/4 = ( m –8 ) –3/4 ( n 2/3 ) –3/4 ( m 1/4 ) –3/4 ( n –6 ) –3/4 = m 6 n –1/2 m –3/16 n 9/2 = m –3/16 – 6 n 9/2 – (–1/2) Quotient rule = m –99/16 n 5 Definition of Negative exponent = m 99/16 n5n5 Product to Power

MTH55_Lec-40_sec_7-2b_Rational_Exponents.ppt 8 Bruce Mayer, PE Chabot College Mathematics Example  Laws of Exponents  Write with only positive exponents. All variables represent positive numbers x 3/5 ( x –1/2 – x 3/4 ) = x 3/5 · x –1/2 – x 3/5 · x 3/4 Distributive property = x 3/5 + (–1/2) – x 3/5 + 3/4 Product rule = x 1/10 – x 27/20  Do not make the common mistake of multiplying exponents in the first step.

MTH55_Lec-40_sec_7-2b_Rational_Exponents.ppt 9 Bruce Mayer, PE Chabot College Mathematics Simplifying Radical Expressions  Many radical expressions contain radicands or factors of radicands that are powers.  When these powers and the index share a common factor, rational exponents can be used to simplify the radical expression.

MTH55_Lec-40_sec_7-2b_Rational_Exponents.ppt 10 Bruce Mayer, PE Chabot College Mathematics Simplifying Radical Expressions 1.Convert radical expressions to exponential expressions. 2.Use arithmetic and the laws of exponents to simplify. 3.Convert back to radical notation when appropriate.  CAUTION  CAUTION: This procedure works only when all expressions under radicals are nonnegative since rational exponents are not defined otherwise. With this assumption, no absolute-value signs will be needed.

MTH55_Lec-40_sec_7-2b_Rational_Exponents.ppt 11 Bruce Mayer, PE Chabot College Mathematics Example  Radical Exponents  Use rational exponents to simplify. a.b.  SOLUTION a. b.

MTH55_Lec-40_sec_7-2b_Rational_Exponents.ppt 12 Bruce Mayer, PE Chabot College Mathematics Example  Radical Exponents  Use rational exponents to simplify. Do not use exponents that are fractions in the final answer.  SOLUTION Convert to exponential notation Simplify the exponent and return to radical notation

MTH55_Lec-40_sec_7-2b_Rational_Exponents.ppt 13 Bruce Mayer, PE Chabot College Mathematics Example  Radical Exponents  SOLUTION

MTH55_Lec-40_sec_7-2b_Rational_Exponents.ppt 14 Bruce Mayer, PE Chabot College Mathematics Example  Radical Exponents  Write a single radical expression for  SOLN

MTH55_Lec-40_sec_7-2b_Rational_Exponents.ppt 15 Bruce Mayer, PE Chabot College Mathematics Rules of Exponents Summary AAssume that no denominators are 0, that a and b are real numbers, and that m and n are integers. ZZero as an exponent:a 0 = 1, where a ≠ is indeterminate. NNegative exponents: PProduct rule for exponents: QQuotient rule for exponents: RRaising a power to a power: RRaising a product to a power: RRaising a quotient to a power:

MTH55_Lec-40_sec_7-2b_Rational_Exponents.ppt 16 Bruce Mayer, PE Chabot College Mathematics Simplification GuideLines  The GuideLines for Simplifying expressions with Rational Exponents 1.No parentheses appear 2.No powers are raised to powers 3.Each Base Occurs only Once 4.No negative or zero exponents appear

MTH55_Lec-40_sec_7-2b_Rational_Exponents.ppt 17 Bruce Mayer, PE Chabot College Mathematics Example  Use Exponent Rules  Rewrite all radicals as exponentials, and then apply the rules for rational exponents. Leave answers in exponential form. Assume c > 0 Convert to rational exponents. Quotient rule Write exponents with a common denominator 4 c c3c3 = c 1/4 c 3/2 = c 1/4 – 3/2 = c 1/4 – 6/4 = c –5/4 = c 5/4 1 Definition of negative exponent

MTH55_Lec-40_sec_7-2b_Rational_Exponents.ppt 18 Bruce Mayer, PE Chabot College Mathematics WhiteBoard Work  Problems From §7.2 Exercise Set 58, 74, 78, 106, 110, 112, 132  America’s Cup “Class Rule” 5.0 Formula

MTH55_Lec-40_sec_7-2b_Rational_Exponents.ppt 19 Bruce Mayer, PE Chabot College Mathematics All Done for Today Radical Index Radicand

MTH55_Lec-40_sec_7-2b_Rational_Exponents.ppt 20 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer Chabot Mathematics Appendix –

MTH55_Lec-40_sec_7-2b_Rational_Exponents.ppt 21 Bruce Mayer, PE Chabot College Mathematics Graph y = |x|  Make T-table

MTH55_Lec-40_sec_7-2b_Rational_Exponents.ppt 22 Bruce Mayer, PE Chabot College Mathematics