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Properties of Exponents

Properties of Exponents. Lesson 2.2. Objectives: Evaluate expressions involving exponents Simplify expressions involving exponents. Some amusement park rides travel in a circle at high speeds. The centripetal acceleration acting on the rider can be calculated with a formula.

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Properties of Exponents

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  1. Properties of Exponents Lesson 2.2

  2. Objectives: • Evaluate expressions involving exponents • Simplify expressions involving exponents

  3. Some amusement park rides travel in a circle at high speeds. The centripetal acceleration acting on the rider can be calculated with a formula. is the centripetal acceleration in feet per second squared. is the radius of the circle in feet is the time for a full rotation in seconds.

  4. Notice that this formula has exponents in it. is called apowerof is called thebase is called theexponent

  5. Exploring Integer Exponents with the Graphing Calculator Go to y= and enter 2^x. Go to TBLSET and set TblStart at -4 and Tbl = 1. Go to TABLE and complete the chart. 2-4= 2-3= 2-2= 2-1= 1/16 1/8 1/4 1/2 21= 22= 23= 24= 2 4 8 16 20= 1

  6. Exploring Integer Exponents with the Graphing Calculator Go to y= and enter 3^x. Go to TABLE and complete the chart. 3-4= 3-3= 3-2= 3-1= 1/81 1/27 1/9 1/3 31= 32= 33= 34= 3 9 27 81 30= 1

  7. Exploring Integer Exponents with the Graphing Calculator Go to y= and enter 4^x. Go to TABLE and complete the chart. 4-4= 4-3= 4-2= 4-1= 1/256 1/64 1/16 1/4 41= 42= 43= 44= 4 16 64 256 40= 1

  8. Using your results, make a conjecture about a-n

  9. Definition of Integer Exponents Let a be a real number. If n is a natural number, then an = a x a x a . . . x a, n times. If a is nonzero*, then a0 = 1 If n is a natural number, then a-n = 1/an *In the expression a0, a must be nonzero because 00 is undefined.

  10. Find the centripetal acceleration in feet per second squared of a rider who makes one rotation in 2 seconds and whose radius of rotation is 6 feet. The centripetal acceleration is about 59 feet per second squared.

  11. Find the centripetal acceleration in feet per second squared of a rider who makes one rotation in 5 seconds and whose radius of rotation is 6 feet. The centripetal acceleration is about 9.5 ft/s2

  12. Find the centripetal acceleration in feet per second squared of a rider who makes one rotation in 4.5 seconds and whose radius of rotation is 8 feet. The centripetal acceleration is about 15.6 ft/s2

  13. Activity Exploring Properties of Exponents • Rewrite (a3)(a5)by writing out all of the factors of a, counting them, and simplifying them as a power with a single exponent. What operation could you perform on the exponents (a3)(a5) to obtain an equivalent expression with a single exponent?

  14. (a3)(a5) = (a x a x a)(a x a x a x a x a) = a8 You can add the exponents Product of Powers Property of Exponents Let a and b be nonzero real numbers. Let m and n be integers. (a)m(a)n = am+n

  15. Activity Exploring Properties of Exponents • Rewrite (a3)5by writing out five sets of three factors of a, counting the factors of a, and simplifying them as a power with a single exponent. What operation could you perform on the exponents in (a3)5 to obtain an equivalent expression with a single exponent?

  16. (a3)5 = (ax a x a)(a x ax a)(a x a x a)(a x a x a)(a x a x a) = a15 You can multiply the exponents Power of a Power Property of Exponents Let a and b be nonzero real numbers. Let m and n be integers. (am)n = amn

  17. Activity Exploring Properties of Exponents • Explain how to simplify (a7 x a3)2 by using addition and multiplication First add the exponents inside the parentheses: a7 x a3 = a10 Then multiply the resulting exponent by 2: (a10)2 = a20

  18. Activity Exploring Properties of Exponents • Rewrite (a5)/(a2)by writing out all of the factors of a, and canceling common factors to simplify the fraction. What operation could you perform on the exponents (a5)/(a 2) to obtain an equivalent expression with a single exponent?

  19. You can subtract the exponents Quotient of Powers Property of Exponents Let a and b be nonzero real numbers. Let m and n be integers.

  20. Two other Properties Power of a Product Property of Exponents Let a and b be nonzero real numbers. Let n be an integer. (ab)n = anbn Power of a Quotient Property of Exponents Let a and b be nonzero real numbers. Let n be an integer.

  21. Applying the Properties Simplify 3x2y-2(-2x3y-4) Write the answer with positive exponents. (3)(-2)(x2x3y-2y-4)Commutative Property Product of Powers Property (3)(-2)(x(2+3)y(-2+(-4))) -6x5y-6Simplify Use a-n = 1/an

  22. Simplify 2z(3x2)(5z-3) (2)(3)(5)(x2z-2) = Simplify 9a2b3(-2a5b-3)2 = 9a2b3(4a10b-6) =

  23. Powers with a Negative Base Look for a pattern: (-2)2 = (-2)(-2) = 4 (-2)3 = (-2)(-2)(-2) = -8 (-2)4 = (-2)(-2)(-2)(-2) = 16 (-2)5 = (-2)(-2)(-2)(-2)(-2) = -32 When the exponent of a negative base is even, the result is positive. When the exponent of a negative base is odd, the result is negative.

  24. A WARNING!!! Do not confuse the results of a negative base with those of a negative exponent. Even Exponent Odd Exponent

  25. Write your answer with positive exponents only. Power of a Quotient Property Power of a Power Property Quotient of Powers Property

  26. Write your answer with positive exponents only. Write your answer with positive exponents only.

  27. Homework: p. 99 (19 - 30, 39 - 58 odd)

  28. Rational Exponents An expression with rational exponents can be represented in an equivalent form that involves the radical symbol,  . Definition of Rational Exponents For all positive real numbers a: If a is a nonzero integer, then If m and n are integers and n0, then

  29. Consider the following:

  30. Using the Definition of Rational Expressions to Evaluate Expressions Evaluate 161/4 161/4 = (24)1/4 = 2(4)(1/4) = 21 = 2 This evaluation can be made two ways on the calculator as shown.

  31. Using the Definition of Rational Expressions to Evaluate Expressions Evaluate 274/3 274/3 = (33)4/3 = 3(3)(4/3) = 34 = 81 This evaluation can be made two ways on the calculator as shown.

  32. Evaluate each expression: 641/3 4 363/2 216 1251/3 5 82/3 4

  33. Application A formula can be used to estimate a person’s surface area based on his or her weight and height. This formula is used to calculate dosages for certain medications. S = 0.007184W0.425H0.725 Sis the surface area in square meters Wis the weight in kilograms His the height in centimeters

  34. Application Estimate to the nearest tenth of a square meter the surface of a person who stands 152.5 cm tall and weighs 57.2 kg. S = 0.007184W0.425H0.725 S = 0.00718(57.2)0.425(152.5)0.725  1.54 m2

  35. Application Estimate to the nearest tenth of a square meter the surface of a person who stands 180 cm tall and weighs 62.3 kg. S = 0.007184W0.425H0.725 S = 0.00718(62.3)0.425(180)0.725  1.8 m2

  36. Homework: p. 99 (31 - 38, 40 - 58 even

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