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5.7 – Rational Exponents

5.7 – Rational Exponents. Basic concept Converting from rational exponent to radical form Evaluating expressions with rational exponents Simplifying. Writing radicals in exponential form. Note that ( √x) 2 = x = x 1 Is there a way for us to represent √x using exponents?

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5.7 – Rational Exponents

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  1. 5.7 – Rational Exponents Basic concept Converting from rational exponent to radical form Evaluating expressions with rational exponents Simplifying

  2. Writing radicals in exponential form • Note that (√x)2 = x = x1 • Is there a way for us to represent √x using exponents? • Suppose we call the power equivalent to a square root “z” • Then (√x)2 = (xz)2 = x = x1 • By the third law of exponents, (xz)2 = x2z • Since x2z = x1, 2z = 1 and z = ½ • Therefore x1/2 is equivalent to √x

  3. More on rational exponents • By similar reasoning, • 3√2 is equivalent to 21/3 • 4√5xy is equivalent to (5xy)1/4 • What about something like 5√x3? • This is the 5th root of x cubed • To write using rational exponents, the index goes in the denominator, and the power goes in the numerator • Thus, 5√x3 = x 3/5 • Similarly, √28 = 28/2 = 24 = 16 • Note from the above example that rational exponents may be reduced like any other fraction

  4. Write in radical form. Definition of Answer: Example 7-1a

  5. Write in radical form. Definition of Answer: Example 7-1b

  6. Write each expression in radical form. a. b. Answer: Answer: Example 7-1c

  7. Write using rational exponents. Answer: Definition of Example 7-2a

  8. Write using rational exponents. Answer: Definition of Example 7-2b

  9. Write each radical using rational exponents. a. b. Answer: Answer: Example 7-2c

  10. Evaluting expressions with radical exponents • Remember, for rational exponents, the number in the numerator is an exponent while a number in a denominator is a root • Usually you want to deal with the root first, then the power that was in the numerator • Also recall that a negative exponent may be made positive by….? • Switching the expression from numerator to denominator or vice versa • See example on the next slide

  11. Evaluate Method 1 Answer: Simplify. Example 7-3a

  12. Method 2 Power of a Power Multiply exponents. Answer: Example 7-3b

  13. Evaluate . Method 1 Factor. Power of a Power Expand the square. Find the fifth root. Example 7-3c Answer: The root is 4.

  14. Method 2 Power of a Power Multiply exponents. Example 7-3d Answer: The root is 4.

  15. Evaluate each expression. a. b. Answer: Example 7-3e Answer: 8

  16. Weight LiftingThe formula can be used toestimatethe maximum total mass that a weight lifter of mass B kilograms can lift in two lifts, the snatch and theclean and jerk, combined. Original formula Use a calculator. Example 7-4a According to theformula, what is the maximum that U.S. WeightlifterOscar Chaplin III can lift if he weighs 77 kilograms? Answer:The formula predicts that he can lift at most 372 kg.

  17. Weight LiftingThe formula can be used toestimatethe maximum total mass that a weight lifter of mass B kilograms can lift in two lifts, the snatch and theclean and jerk, combined. Example 7-4b Oscar Chaplin’s total in the 2000 Olympics was 355 kg. Compare this to the value predicted by the formula. Answer:The formula prediction is somewhat higher than his actual total.

  18. Weight LiftingUse the formula where M is the maximum total mass that a weight lifter of mass B kilograms canlift. a. According to the formula, what is the maximum that a weight lifter can lift if he weighs 80 kilograms? b. If he actually lifted 379 kg, compare this to the value predicted by the formula. Example 7-4c Answer:380 kg Answer:The formula prediction is slightly higher than his actual total.

  19. Laws of Exponents • The laws of exponents hold true for expressions with rational exponents • That is, the following still hold true • xa * xb = xa+b • xa÷ xb = xa – b • (xa)b = xab • Recall that for the 1st two laws, the base of each expression must be the same • Also remember that when combining fractions you often need to find a common denominator • Remember – we don’t like to have negative exponents in our answers and we don’t like having radicals in denominators (sometimes necessitating rationalizing a denominator) • Finally, if possible, you should reduce the index so it is a small as possible • Let’s look at a few examples!

  20. Simplify . Multiply powers. Answer: Add exponents. Example 7-5a

  21. Simplify . Multiply by Example 7-5b

  22. Answer: Example 7-5c

  23. Simplify each expression. a. b. Answer: Answer: Example 7-5d

  24. Simplify . Rational exponents Power of a Power Example 7-6a

  25. Quotient of Powers Answer: Simplify. Example 7-6b

  26. Simplify . Rational exponents Power of a Power Multiply. Answer: Simplify. Example 7-6c

  27. Simplify . is the conjugate of Answer: Multiply. Example 7-6d

  28. Simplify each expression. a. b. c. Answer: Answer: Example 7-6e Answer: 1

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